Stochastic Heat Equation Benchmarks

Chris Rackauckas

Stochastic Heat Equation Benchmarks

In this notebook we will benchmark against the stochastic heat equation with Dirichlet BCs and scalar noise. The function for generating the problem is as follows:

Stochastic Heat Equation with scalar multiplicative noise

S-ROCK: CHEBYSHEV METHODS FOR STIFF STOCHASTIC DIFFERENTIAL EQUATIONS

ASSYR ABDULLE AND STEPHANE CIRILLI

Raising D or k increases stiffness

using StochasticDiffEq, DiffEqNoiseProcess, LinearAlgebra, Statistics

function generate_stiff_stoch_heat(D=1,k=1;N = 100, t_end = 3.0, adaptivealg = :RSwM3)
    A = Array(Tridiagonal([1.0 for i in 1:N-1],[-2.0 for i in 1:N],[1.0 for i in 1:N-1]))
    dx = 1/N
    A = D/(dx^2) * A
    function f(du,u,p,t)
        mul!(du,A,u)
    end
    #=
    function f(::Type{Val{:analytic}},u0,p,t,W)
        exp((A-k/2)*t+W*I)*u0 # no -k/2 for Strat
    end
    =#
    function g(du,u,p,t)
        @. du = k*u
    end
    SDEProblem(f,g,ones(N),(0.0,t_end),noise=WienerProcess(0.0,0.0,0.0,rswm=RSWM(adaptivealg=adaptivealg)))
end

N = 100
D = 1; k = 1
    A = Array(Tridiagonal([1.0 for i in 1:N-1],[-2.0 for i in 1:N],[1.0 for i in 1:N-1]))
    dx = 1/N
    A = D/(dx^2) * A;
100×100 Array{Float64,2}:
 -20000.0   10000.0       0.0       0.0  …       0.0       0.0       0.0
  10000.0  -20000.0   10000.0       0.0          0.0       0.0       0.0
      0.0   10000.0  -20000.0   10000.0          0.0       0.0       0.0
      0.0       0.0   10000.0  -20000.0          0.0       0.0       0.0
      0.0       0.0       0.0   10000.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0  …       0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      ⋮                                  ⋱                      
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0          0.0       0.0       0.0
      0.0       0.0       0.0       0.0  …       0.0       0.0       0.0
      0.0       0.0       0.0       0.0      10000.0       0.0       0.0
      0.0       0.0       0.0       0.0     -20000.0   10000.0       0.0
      0.0       0.0       0.0       0.0      10000.0  -20000.0   10000.0
      0.0       0.0       0.0       0.0          0.0   10000.0  -20000.0

Now lets solve it with high accuracy.

prob = generate_stiff_stoch_heat(1.0,1.0)
@time sol = solve(prob,SRIW1(),progress=true,abstol=1e-6,reltol=1e-6);
21.384544 seconds (48.69 M allocations: 8.744 GiB, 5.28% gc time)
retcode: Success
Interpolation: 1st order linear
t: 640974-element Array{Float64,1}:
 0.0
 4.047220023712516e-9
 4.8566640284550185e-9
 5.7672885337903344e-9
 6.791741102292565e-9
 7.944250241857574e-9
 9.24082302386821e-9
 1.0674357215215282e-8
 1.2234913360521535e-8
 1.3911888496432656e-8
 ⋮
 2.99937072137815
 2.9994221941199983
 2.9994801009545777
 2.9995452461434797
 2.9996185344809945
 2.9997009838606985
 2.9997937394128655
 2.9998980894090534
 3.0
u: 640974-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [1.0000772461210554, 1.0001177206285683, 1.0001177214475678, 1.00011772144
75678, 1.0001177214475678, 1.0001177214475678, 1.0001177214475678, 1.000117
7214475678, 1.0001177214475678, 1.0001177214475678  …  1.0001177214475678, 
1.0001177214475678, 1.0001177214475678, 1.0001177214475678, 1.0001177214475
678, 1.0001177214475678, 1.0001177214475678, 1.0001177214475678, 1.00011772
06285683, 1.0000772461210554]
 [1.000060397550535, 1.000108965944716, 1.0001089671240677, 1.0001089671240
757, 1.0001089671240757, 1.0001089671240757, 1.0001089671240757, 1.00010896
71240757, 1.0001089671240757, 1.0001089671240757  …  1.0001089671240757, 1.
0001089671240757, 1.0001089671240757, 1.0001089671240757, 1.000108967124075
7, 1.0001089671240757, 1.0001089671240757, 1.0001089671240677, 1.0001089659
44716, 1.000060397550535]
 [1.0000520920722988, 1.000109766298793, 1.0001097679618565, 1.000109767961
8771, 1.0001097679618771, 1.0001097679618771, 1.0001097679618771, 1.0001097
679618771, 1.0001097679618771, 1.0001097679618771  …  1.0001097679618771, 1
.0001097679618771, 1.0001097679618771, 1.0001097679618771, 1.00010976796187
71, 1.0001097679618771, 1.0001097679618771, 1.0001097679618565, 1.000109766
298793, 1.0000520920722988]
 [1.0000656255469498, 1.0001335451085756, 1.0001335474149693, 1.00013354741
501, 1.00013354741501, 1.00013354741501, 1.00013354741501, 1.00013354741501
, 1.00013354741501, 1.00013354741501  …  1.00013354741501, 1.00013354741501
, 1.00013354741501, 1.00013354741501, 1.00013354741501, 1.00013354741501, 1
.00013354741501, 1.0001335474149693, 1.0001335451085756, 1.0000656255469498
]
 [1.0000265551886822, 1.0001059966450963, 1.0001059998005308, 1.00010599980
06026, 1.0001059998006026, 1.0001059998006026, 1.0001059998006026, 1.000105
9998006026, 1.0001059998006026, 1.0001059998006026  …  1.0001059998006026, 
1.0001059998006026, 1.0001059998006026, 1.0001059998006026, 1.0001059998006
026, 1.0001059998006026, 1.0001059998006026, 1.0001059998005308, 1.00010599
66450963, 1.0000265551886822]
 [1.0000473842254047, 1.0001397925645164, 1.0001397968340575, 1.00013979683
41768, 1.0001397968341768, 1.0001397968341768, 1.0001397968341768, 1.000139
7968341768, 1.0001397968341768, 1.0001397968341768  …  1.0001397968341768, 
1.0001397968341768, 1.0001397968341768, 1.0001397968341768, 1.0001397968341
768, 1.0001397968341768, 1.0001397968341768, 1.0001397968340575, 1.00013979
25645164, 1.0000473842254047]
 [1.0000192485733945, 1.0001259885030316, 1.0001259941997966, 1.00012599419
99865, 1.0001259941999865, 1.0001259941999865, 1.0001259941999865, 1.000125
9941999865, 1.0001259941999865, 1.0001259941999865  …  1.0001259941999865, 
1.0001259941999865, 1.0001259941999865, 1.0001259941999865, 1.0001259941999
865, 1.0001259941999865, 1.0001259941999865, 1.0001259941997966, 1.00012598
85030316, 1.0000192485733945]
 [0.9999299231487992, 1.0000522562243923, 1.0000522637078906, 1.00005226370
81821, 1.0000522637081821, 1.0000522637081821, 1.0000522637081821, 1.000052
2637081821, 1.0000522637081821, 1.0000522637081821  …  1.0000522637081821, 
1.0000522637081821, 1.0000522637081821, 1.0000522637081821, 1.0000522637081
821, 1.0000522637081821, 1.0000522637081821, 1.0000522637078906, 1.00005225
62243923, 0.9999299231487992]
 [0.9999164814411606, 1.0000555790308598, 1.0000555887061677, 1.00005558870
6602, 1.000055588706602, 1.000055588706602, 1.000055588706602, 1.0000555887
06602, 1.000055588706602, 1.000055588706602  …  1.000055588706602, 1.000055
588706602, 1.000055588706602, 1.000055588706602, 1.000055588706602, 1.00005
5588706602, 1.000055588706602, 1.0000555887061677, 1.0000555790308598, 0.99
99164814411606]
 ⋮
 [3.5108775980116924e-11, -7.014478846678073e-11, 1.0519049890654109e-10, -
1.4001820037386725e-10, 1.7486526022770012e-10, -2.0934990906465644e-10, 2.
4386350182772903e-10, -2.7787168362684834e-10, 3.1191828291419223e-10, -3.4
53184266261204e-10  …  3.4550942686279973e-10, -3.1174585211756133e-10, 2.7
80253781678169e-10, -2.437286922318809e-10, 2.094657032959323e-10, -1.74768
59338430288e-10, 1.40095646310584e-10, -1.0513234880034912e-10, 7.018358648
604713e-11, -3.508936758230944e-11]
 [3.743194250603354e-11, -7.47891180255601e-11, 1.121510106043199e-10, -1.4
928889148259094e-10, 1.864361882298071e-10, -2.2321109522038975e-10, 2.6000
008046158185e-10, -2.9626973859315244e-10, 3.325580828918209e-10, -3.681821
7190251186e-10  …  3.683719708414327e-10, -3.323867366002612e-10, 2.9642246
647092474e-10, -2.5986611875187605e-10, 2.2332616116284146e-10, -1.86340129
37335538e-10, 1.4936585032242135e-10, -1.1209322623372319e-10, 7.4827672024
34832e-11, -3.7412656177512363e-11]
 [5.1383951794078816e-11, -1.0267987602228782e-10, 1.5395306085494362e-10, 
-2.0496250355160168e-10, 2.5592655543137854e-10, -3.064521709721249e-10, 3.
5690991988309877e-10, -4.067562344765602e-10, 4.5651246919092477e-10, -5.05
4866371289613e-10  …  5.056752688957918e-10, -4.563421765965353e-10, 4.0690
802315146493e-10, -3.5677678197334674e-10, 3.0656652931450075e-10, -2.55831
08729057893e-10, 2.0503898913167578e-10, -1.538956318303634e-10, 1.02718192
93257319e-10, -5.136478406717896e-11]
 [9.197967888151542e-11, -1.8383212821306003e-10, 2.7558318513557683e-10, -
3.6695304563462545e-10, 4.5812051358312805e-10, -5.486542930095137e-10, 6.3
88854627615265e-10, -7.282329035100967e-10, 8.171786881991963e-10, -9.04994
1223785605e-10  …  9.051824008753151e-10, -8.170087145292077e-10, 7.2838440
79148138e-10, -6.387525741927445e-10, 5.487684371813517e-10, -4.58025224235
24924e-10, 3.6702938797236034e-10, -2.7552586366418217e-10, 1.8387037336336
404e-10, -9.196054705197073e-11]
 [2.1676481310849923e-10, -4.332815595201299e-10, 6.494558183738685e-10, -8
.648868368588646e-10, 1.0796342052459298e-9, -1.2931460350448628e-9, 1.5056
356971155094e-9, -1.7164023024369573e-9, 1.9258121769687468e-9, -2.13301814
2761408e-9  …  2.1332069902939307e-9, -1.925641689585808e-9, 1.716554264734
8269e-9, -1.5055024069163677e-9, 1.293260524195715e-9, -1.0795386279041415e
-9, 8.64963409926786e-10, -6.493983236592354e-10, 4.3331992025912427e-10, -
2.1674562345664026e-10]
 [6.856319904026236e-10, -1.3705627574233617e-9, 2.0542433942328058e-9, -2.
735823073793613e-9, 3.414907329442826e-9, -4.090499022171944e-9, 4.76235965
1599785e-9, -5.429349624483096e-9, 6.091387329435081e-9, -6.747195127841479
e-9  …  6.74738179929913e-9, -6.091218806564483e-9, 5.429499835731726e-9, -
4.762227897292795e-9, 4.090612192073629e-9, -3.4148128534309675e-9, 2.73589
87645163212e-9, -2.054186562025081e-9, 1.3706006761346943e-9, -6.8561302187
15572e-10]
 [2.8573655214541785e-9, -5.7119288061408745e-9, 8.561041972252408e-9, -1.1
401759269371116e-8, 1.423159741478959e-8, -1.704747854097581e-8, 1.98470935
7166147e-8, -2.2627244429871077e-8, 2.53858051791438e-8, -2.811946990579823
e-8  …  2.8119656584411337e-8, -2.5385636649813458e-8, 2.262739464687763e-8
, -1.984696181230409e-8, 1.704759171521559e-8, -1.4231502935156253e-8, 1.14
01834962995265e-8, -8.560985137866171e-9, 5.711966726305744e-9, -2.85734655
2196003e-9]
 [1.5896051007440475e-8, -3.177668608223374e-8, 4.762665428905479e-8, -6.34
3043433918277e-8, 7.917299929752102e-8, -9.483878256812556e-8, 1.1041303935
242123e-7, -1.2588021797137583e-7, 1.4122591280769782e-7, -1.56434647265690
9e-7  …  1.5643483202318775e-7, -1.4122574601270918e-7, 1.258803666424939e-
7, -1.1041290894900439e-7, 9.483889457768528e-8, -7.917290579018116e-8, 6.3
43050925384122e-8, -4.762659803955335e-8, 3.17767236121609e-8, -1.589603223
339565e-8]
 [8.24538082636906e-8, -1.6482780978317076e-7, 2.4704242702496914e-7, -3.29
0179327909257e-7, 4.106752866283259e-7, -4.919351493341012e-7, 5.7271932332
84509e-7, -6.529491643513613e-7, 7.325476202271914e-7, -8.114370458983448e-
7  …  8.114372330187664e-7, -7.32547451299009e-7, 6.529493149238836e-7, -5.
727191912572636e-7, 4.919352627761882e-7, -4.10675191925091e-7, 3.290180086
636925e-7, -2.47042370056075e-7, 1.6482784779308065e-7, -8.245378924953859e
-8]

Highest dt

Let's try to find the highest possible dt:

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),SRIW1());
1.472808 seconds (1.01 M allocations: 131.921 MiB, 8.83% gc time)
retcode: Success
Interpolation: 1st order linear
t: 93553-element Array{Float64,1}:
 0.0
 1.535027855349674e-5
 1.842033426419609e-5
 2.1874146938732856e-5
 2.575968619758672e-5
 3.0130917863797315e-5
 3.504855348828424e-5
 4.0580893565832026e-5
 4.6804776153073284e-5
 5.38066440637197e-5
 ⋮
 2.999656804878185
 2.999685163485566
 2.9997170669188695
 2.999752958281336
 2.9997933360641107
 2.9998387610697326
 2.999889864201057
 2.999947355223797
 3.0
u: 93553-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.8689437129204255, 0.9869011511361877, 0.9986827037196847, 0.99868270371
96847, 0.9986827037196847, 0.9986827037196847, 0.9986827037196847, 0.998682
7037196847, 0.9986827037196847, 0.9986827037196847  …  0.9986827037196847, 
0.9986827037196847, 0.9986827037196847, 0.9986827037196847, 0.9986827037196
847, 0.9986827037196847, 0.9986827037196847, 0.9986827037196847, 0.98690115
11361877, 0.8689437129204255]
 [0.8492579689291411, 0.986535872118059, 1.0014828975818229, 1.001879136413
7047, 1.0018846886129558, 1.0018846886129558, 1.0018846886129558, 1.0018846
886129558, 1.0018846886129558, 1.0018846886129558  …  1.0018846886129558, 1
.0018846886129558, 1.0018846886129558, 1.0018846886129558, 1.00188468861295
58, 1.0018846886129558, 1.0018791364137047, 1.0014828975818229, 0.986535872
118059, 0.8492579689291411]
 [0.8257339701113886, 0.9823689896299493, 1.0012766785033238, 1.00220991561
23947, 1.0022367658408897, 1.0022371807580461, 1.002237184069606, 1.0022371
84069606, 1.002237184069606, 1.002237184069606  …  1.002237184069606, 1.002
237184069606, 1.002237184069606, 1.002237184069606, 1.0022371807580461, 1.0
022367658408897, 1.0022099156123947, 1.0012766785033238, 0.9823689896299493
, 0.8257339701113886]
 [0.7988394191582092, 0.9745045953183106, 0.9982454325418794, 0.99990355210
71145, 0.9999760309725433, 0.9999780791064193, 0.9999781172698126, 0.999978
1177014043, 0.9999781177039041, 0.9999781177039041  …  0.9999781177039041, 
0.9999781177039041, 0.9999781177014043, 0.9999781172698126, 0.9999780791064
193, 0.9999760309725433, 0.9999035521071145, 0.9982454325418794, 0.97450459
53183106, 0.7988394191582092]
 [0.7706911665046717, 0.965087132571645, 0.9946508883606457, 0.997284848094
7304, 0.9974399618885034, 0.9974463068517769, 0.9974464925466773, 0.9974464
964170119, 0.9974464964729312, 0.997446496473443  …  0.997446496473443, 0.9
974464964729312, 0.9974464964170119, 0.9974464925466773, 0.9974463068517769
, 0.9974399618885034, 0.9972848480947304, 0.9946508883606457, 0.96508713257
1645, 0.7706911665046717]
 [0.744401307593628, 0.9577538566943191, 0.9943346544096191, 0.998285879351
526, 0.9985799914520521, 0.9985958446194544, 0.9985964830136232, 0.99859650
24541353, 0.9985965029014048, 0.9985965029090331  …  0.9985965029090331, 0.
9985965029014048, 0.9985965024541353, 0.9985964830136232, 0.998595844619454
4, 0.9985799914520521, 0.998285879351526, 0.9943346544096191, 0.95775385669
43191, 0.744401307593628]
 [0.7153815860130207, 0.9463334061411511, 0.9909719928300206, 0.99665543121
85185, 0.997170202908109, 0.9972049543481287, 0.99720675701531, 0.997206830
1032119, 0.9972068324369, 0.9972068324954723  …  0.9972068324954723, 0.9972
068324369, 0.9972068301032119, 0.99720675701531, 0.9972049543481287, 0.9971
70202908109, 0.9966554312185185, 0.9909719928300206, 0.9463334061411511, 0.
7153815860130207]
 [0.6848440754493929, 0.931849853842847, 0.9855824540361463, 0.993510717691
6478, 0.9943632084785425, 0.9944330212029695, 0.9944375047023004, 0.9944377
347174062, 0.9944377442467467, 0.9944377445669506  …  0.9944377445669506, 0
.9944377442467467, 0.9944377347174062, 0.9944375047023005, 0.99443302120296
95, 0.9943632084785425, 0.9935107176916478, 0.9855824540361462, 0.931849853
842847, 0.6848440754493929]
 [0.65853516008368, 0.9218341922636543, 0.9861256925698435, 0.9969962156938
239, 0.9983604669578648, 0.9984930571752603, 0.9985033218080627, 0.99850396
67629386, 0.9985040000570758, 0.9985040014791589  …  0.9985040014791589, 0.
9985040000570758, 0.9985039667629386, 0.9985033218080628, 0.998493057175260
3, 0.9983604669578648, 0.9969962156938239, 0.9861256925698434, 0.9218341922
636543, 0.65853516008368]
 ⋮
 [-3.8012970434908513e-6, 7.5989165820009965e-6, -1.1389184650686386e-5, 1.
5168434431975585e-5, -1.8933009715037527e-5, 2.2679268555880243e-5, -2.6403
586642901386e-5, 3.0102360995612283e-5, -3.377201322343735e-5, 3.7408993248
60942e-5  …  -3.7408993237194184e-5, 3.3772013252248055e-5, -3.010236098657
558e-5, 2.6403586665547855e-5, -2.267926854915051e-5, 1.8933009731347343e-5
, -1.5168434427509782e-5, 1.138918466052775e-5, -7.598916579773426e-6, 3.80
1297046780718e-6]
 [-1.937384106171051e-6, 3.87289392272423e-6, -5.804656955781333e-6, 7.7308
04371240683e-6, -9.64947271529886e-6, 1.1558805839492876e-5, -1.34569565394
78675e-5, 1.534208853346373e-5, -1.7212378013073565e-5, 1.906601566739512e-
5  …  -1.9066015651092608e-5, 1.721237803721947e-5, -1.5342088520421736e-5,
 1.345695655842035e-5, -1.1558805829703684e-5, 9.649472728920882e-6, -7.730
804364708365e-6, 5.804656963993213e-6, -3.872893919455424e-6, 1.93738410891
49716e-6]
 [-1.0469750782405416e-6, 2.0929372800520587e-6, -3.1368746915401666e-6, 4.
1777773923567705e-6, -5.21463834606015e-6, 6.246454495536586e-6, -7.2722275
7786629e-6, 8.290965279761357e-6, -9.301681973939468e-6, 1.0303399930001675
e-5  …  -1.0303399911497329e-5, 9.301681995736634e-6, -8.290965264912922e-6
, 7.272227594943698e-6, -6.2464544843669e-6, 5.214638358330082e-6, -4.17777
7384891703e-6, 3.136874698932491e-6, -2.092937276313218e-6, 1.0469750807098
938e-6]
 [-6.211320582413286e-7, 1.2416632156853451e-6, -1.8609931354588935e-6, 2.4
785226776024052e-6, -3.093654391773026e-6, 3.705793216774164e-6, -4.3143469
002507605e-6, 4.918726761907833e-6, -5.518348038712815e-6, 6.11263070876161
1e-6  …  -6.112630689047837e-6, 5.518348059527847e-6, -4.9187267460696365e-
6, 4.314346916546653e-6, -3.7057932048498257e-6, 3.0936544034749437e-6, -2.
4785226696282893e-6, 1.8609931425064397e-6, -1.2416632116902378e-6, 6.21132
0605950896e-7]
 [-4.295334652430111e-7, 8.586513889321556e-7, -1.2869386141985882e-6, 1.71
39808224465965e-6, -2.139364847672265e-6, 2.562679197127205e-6, -2.98351429
41662775e-6, 3.401463063840988e-6, -3.816121103457954e-6, 4.227087330765336
e-6  …  -4.227087310635808e-6, 3.816121123714803e-6, -3.4014630476608194e-6
, 2.9835143100193055e-6, -2.562679184941642e-6, 2.1393648590527705e-6, -1.7
139808142961362e-6, 1.2869386210511948e-6, -8.586513848482537e-7, 4.2953346
75314357e-7]
 [-3.564879674055245e-7, 7.126310600743895e-7, -1.068084719526752e-6, 1.422
505089016364e-6, -1.7755492593677428e-6, 2.126875722113016e-6, -2.476144544
061545e-6, 2.823017886225235e-6, -3.1671601065497903e-6, 3.5082383425660248
e-6  …  -3.5082383221582842e-6, 3.167160126700933e-6, -2.8230178698184103e-
6, 2.4761445598291587e-6, -2.12687570975574e-6, 1.7755492706851308e-6, -1.4
225050807507352e-6, 1.0680847263405854e-6, -7.126310559327236e-7, 3.5648796
96808676e-7]
 [-3.731712638542422e-7, 7.459815129475544e-7, -1.1180700644399004e-6, 1.48
90769683396998e-6, -1.858643269156045e-6, 2.226411474218832e-6, -2.59202574
4143049e-6, 2.9551324266258574e-6, -3.315380175073313e-6, 3.672420545769973
4e-6  …  -3.6724205254615974e-6, 3.3153801952130846e-6, -2.9551324102996997
e-6, 2.5920257599017596e-6, -2.226411461923421e-6, 1.8586432804666135e-6, -
1.4890769601161718e-6, 1.1180700712493708e-6, -7.459815088272325e-7, 3.7317
12661280777e-7]
 [-5.044232509708347e-7, 1.008358509330173e-6, -1.511318237349569e-6, 2.012
81587547788e-6, -2.51236622707321e-6, 3.009486048027592e-6, -3.503694359940
8838e-6, 3.994513103706556e-6, -4.48146737998597e-6, 4.9640861642017665e-6 
 …  -4.964086144383308e-6, 4.4814674003152704e-6, -3.994513087779777e-6, 3.
5036943758512567e-6, -3.009486036036741e-6, 2.5123662384939655e-6, -2.01281
5867460054e-6, 1.5113182442257878e-6, -1.008358505313548e-6, 5.044232532670
163e-7]
 [-5.636681401092879e-7, 1.1267909719742193e-6, -1.6888236977981637e-6, 2.2
492226085759945e-6, -2.8074455246888516e-6, 3.3629524398563573e-6, -3.91520
5890245313e-6, 4.463671661463803e-6, -5.007819084812258e-6, 5.5471218045700
13e-6  …  -5.547121785045092e-6, 5.00781910515894e-6, -4.463671645775572e-6
, 3.915205906170682e-6, -3.3629524280470913e-6, 2.807445536120757e-6, -2.24
9222600680727e-6, 1.6888237046811427e-6, -1.126790968019369e-6, 5.636681424
077252e-7]
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),SRIW1(),progress=true,adaptive=false,dt=0.00005);
0.704713 seconds (606.28 k allocations: 86.552 MiB)
retcode: Success
Interpolation: 1st order linear
t: 60002-element Array{Float64,1}:
 0.0
 5.0e-5
 0.0001
 0.00015000000000000001
 0.0002
 0.00025
 0.00030000000000000003
 0.00035000000000000005
 0.0004000000000000001
 0.0004500000000000001
 ⋮
 2.9996499999996784
 2.9996999999996783
 2.999749999999678
 2.999799999999678
 2.999849999999678
 2.999899999999678
 2.9999499999996777
 2.9999999999996776
 3.0
u: 60002-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.7529236623776787, 0.880835418633156, 1.005835418633156, 1.0058354186331
56, 1.005835418633156, 1.005835418633156, 1.005835418633156, 1.005835418633
156, 1.005835418633156, 1.005835418633156  …  1.005835418633156, 1.00583541
8633156, 1.005835418633156, 1.005835418633156, 1.005835418633156, 1.0058354
18633156, 1.005835418633156, 1.005835418633156, 0.880835418633156, 0.752923
6623776787]
 [0.5954488425905352, 0.7846625950786944, 0.9724029845134884, 0.98827356211
51963, 1.0038985621151963, 1.0038985621151963, 1.0038985621151963, 1.003898
5621151963, 1.0038985621151963, 1.0038985621151963  …  1.0038985621151963, 
1.0038985621151963, 1.0038985621151963, 1.0038985621151963, 1.0038985621151
963, 1.0038985621151963, 0.9882735621151963, 0.9724029845134882, 0.78466259
50786944, 0.5954488425905352]
 [0.4943737873564739, 0.7133721561666058, 0.9307357272002181, 0.96646569884
62593, 1.0016654488101204, 1.0036627607198823, 1.0056158857198823, 1.005615
8857198823, 1.0056158857198823, 1.0056158857198823  …  1.0056158857198823, 
1.0056158857198823, 1.0056158857198823, 1.0056158857198823, 1.0036627607198
823, 1.0016654488101204, 0.9664656988462593, 0.9307357272002179, 0.71337215
61666059, 0.49437378735647386]
 [0.42557118852653497, 0.6563338956630647, 0.8856400848561319, 0.9401519841
953986, 0.9939791773391223, 0.9999567115277221, 1.0058221993959289, 1.00607
25650306548, 1.0063167056556548, 1.0063167056556548  …  1.0063167056556548,
 1.0063167056556548, 1.0060725650306548, 1.0058221993959289, 0.999956711527
722, 0.9939791773391223, 0.9401519841953987, 0.8856400848561315, 0.65633389
56630648, 0.4255711885265349]
 [0.38324940388127654, 0.6228737529665327, 0.8576318809586357, 0.9309353333
415417, 1.0013074405158846, 1.0132317189349627, 1.0244520773041177, 1.02544
80178200567, 1.0263668150621497, 1.0264005470021547  …  1.0264005470021547,
 1.0263668150621497, 1.0254480178200567, 1.0244520773041175, 1.013231718934
9624, 1.0013074405158846, 0.9309353333415417, 0.8576318809586355, 0.6228737
529665328, 0.38324940388127643]
 [0.3462952996786668, 0.582650763807126, 0.815199947082042, 0.9014092213722
937, 0.984874928603517, 1.0030491313769665, 1.0203665471509997, 1.022599881
6453943, 1.024694603861279, 1.024843741979267  …  1.024843741979267, 1.0246
94603861279, 1.0225998816453945, 1.0203665471509995, 1.003049131376966, 0.9
848749286035169, 0.9014092213722938, 0.8151999470820418, 0.582650763807126,
 0.34629529967866673]
 [0.31901324592812197, 0.5510184930430357, 0.7795257392346306, 0.8763682538
812824, 0.9703221856007598, 0.9951260709652442, 1.018832894192176, 1.022821
0386726486, 1.0265767383795616, 1.026970898077588  …  1.026970898077588, 1.
0265767383795616, 1.0228210386726488, 1.0188328941921758, 0.995126070965243
9, 0.9703221856007597, 0.8763682538812824, 0.7795257392346304, 0.5510184930
430359, 0.3190132459281219]
 [0.29613634421366686, 0.52144171322164, 0.7440314209136446, 0.848364935656
576, 0.9502045713147229, 0.9812658829449518, 1.0112235450995462, 1.01733052
22806224, 1.023150864124733, 1.0239426672993586  …  1.0239426672993586, 1.0
23150864124733, 1.0173305222806224, 1.011223545099546, 0.9812658829449515, 
0.9502045713147228, 0.848364935656576, 0.7440314209136444, 0.52144171322164
01, 0.2961363442136668]
 [0.27820516864729034, 0.497359930065635, 0.7141217490573365, 0.82449173199
57045, 0.9324694375555747, 0.9696008078970044, 1.0055346201388025, 1.014104
777545006, 1.0223094564054493, 1.0236763748560784  …  1.0236763748560784, 1
.0223094564054491, 1.0141047775450063, 1.0055346201388022, 0.96960080789700
44, 0.9324694375555744, 0.8244917319957044, 0.7141217490573364, 0.497359930
0656351, 0.2782051686472903]
 ⋮
 [4.336863967278256e-15, 8.669532298760097e-15, 1.2993813417655737e-14, 1.7
305523861261962e-14, 2.160049232819191e-14, 2.587456371384008e-14, 3.012360
3130177224e-14, 3.4343499905988104e-14, 3.853017156367958e-14, 4.2679567768
81502e-14  …  4.267956776881494e-14, 3.85301715636796e-14, 3.43434999059880
5e-14, 3.0123603130177237e-14, 2.5874563713840034e-14, 2.160049232819192e-1
4, 1.7305523861261937e-14, 1.2993813417655742e-14, 8.669532298760089e-15, 4
.336863967278257e-15]
 [4.314428863360102e-15, 8.624683795437883e-15, 1.2926594902959932e-14, 1.7
21600036476429e-14, 2.148875045809346e-14, 2.5740711573183428e-14, 2.996777
021226375e-14, 3.4165836969101586e-14, 3.833085048523729e-14, 4.24587813790
8739e-14  …  4.245878137908731e-14, 3.8330850485237315e-14, 3.4165836969101
53e-14, 2.996777021226377e-14, 2.5740711573183387e-14, 2.148875045809347e-1
4, 1.7216000364764264e-14, 1.2926594902959936e-14, 8.624683795437873e-15, 4
.314428863360104e-15]
 [4.378706692436691e-15, 8.753177268942743e-15, 1.3119179711756069e-14, 1.7
47249019548709e-14, 2.1808897173396828e-14, 2.6124205451796655e-14, 3.04142
40248626844e-14, 3.467485123228917e-14, 3.890191653682389e-14, 4.3091346749
54982e-14  …  4.309134674954974e-14, 3.8901916536823916e-14, 3.467485123228
911e-14, 3.041424024862686e-14, 2.6124205451796617e-14, 2.1808897173396834e
-14, 1.7472490195487066e-14, 1.3119179711756073e-14, 8.753177268942734e-15,
 4.3787066924366915e-15]
 [4.363640830715215e-15, 8.723060120747999e-15, 1.3074040413483843e-14, 1.7
412372416493195e-14, 2.1733859073749778e-14, 2.603431962601162e-14, 3.03095
93655435076e-14, 3.455554511051108e-14, 3.8768066307427056e-14, 4.294308190
398658e-14  …  4.29430819039865e-14, 3.876806630742708e-14, 3.4555545110511
02e-14, 3.0309593655435095e-14, 2.6034319626011574e-14, 2.173385907374979e-
14, 1.7412372416493167e-14, 1.307404041348385e-14, 8.723060120747988e-15, 4
.363640830715217e-15]
 [4.38552990291218e-15, 8.766817088878264e-15, 1.3139622945506738e-14, 1.74
9971706554457e-14, 2.184288133952274e-14, 2.6164914036505133e-14, 3.0461633
86899128e-14, 3.4728884038042735e-14, 3.8962536254716676e-14, 4.31584947339
19825e-14  …  4.315849473391974e-14, 3.89625362547167e-14, 3.47288840380426
65e-14, 3.04616338689913e-14, 2.6164914036505082e-14, 2.1842881339522752e-1
4, 1.7499717065544545e-14, 1.3139622945506744e-14, 8.766817088878251e-15, 4
.385529902912182e-15]
 [4.381394201668692e-15, 8.75854968741514e-15, 1.3127231842000953e-14, 1.74
832142475885e-14, 2.1822282772526936e-14, 2.6140239648252827e-14, 3.0432907
530359545e-14, 3.469613353990892e-14, 3.8925793281072584e-14, 4.31177948312
1943e-14  …  4.3117794831219356e-14, 3.892579328107261e-14, 3.4696133539908
85e-14, 3.0432907530359564e-14, 2.6140239648252783e-14, 2.1822282772526945e
-14, 1.7483214247588476e-14, 1.3127231842000958e-14, 8.758549687415128e-15,
 4.381394201668694e-15]
 [4.3353245154644864e-15, 8.666454884452767e-15, 1.2989201018054431e-14, 1.
729938093856542e-14, 2.1592824825281085e-14, 2.5865379048528868e-14, 3.0112
910188036114e-14, 3.4331309031748024e-14, 3.851649455122402e-14, 4.26644178
4977023e-14  …  4.266441784977015e-14, 3.851649455122406e-14, 3.43313090317
4795e-14, 3.011291018803613e-14, 2.5865379048528817e-14, 2.1592824825281098
e-14, 1.7299380938565394e-14, 1.2989201018054434e-14, 8.666454884452756e-15
, 4.335324515464488e-15]
 [4.3217215469846976e-15, 8.639262107486661e-15, 1.2948444739857503e-14, 1.
7245100588204618e-14, 2.1525072915489752e-14, 2.5784221124904035e-14, 3.001
842476562738e-14, 3.422358751909934e-14, 3.8395641161941786e-14, 4.25305495
0170338e-14  …  4.25305495017033e-14, 3.839564116194182e-14, 3.422358751909
927e-14, 3.001842476562741e-14, 2.5784221124903984e-14, 2.1525072915489765e
-14, 1.72451005882046e-14, 1.2948444739857503e-14, 8.639262107486652e-15, 4
.3217215469847e-15]
 [4.321721173485407e-15, 8.639261360849417e-15, 1.2948443620804629e-14, 1.7
245099097818725e-14, 2.1525071055212692e-14, 2.5784218896535505e-14, 3.0018
422171323184e-14, 3.4223584561369295e-14, 3.839563784364731e-14, 4.25305458
2605471e-14  …  4.253054582605463e-14, 3.839563784364734e-14, 3.42235845613
69226e-14, 3.0018422171323216e-14, 2.5784218896535455e-14, 2.15250710552127
04e-14, 1.7245099097818706e-14, 1.2948443620804629e-14, 8.639261360849408e-
15, 4.321721173485409e-15]
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),EM(),progress=true,adaptive=false,dt=0.00005);
10.089598 seconds (23.88 M allocations: 7.035 GiB, 7.16% gc time)
retcode: Success
Interpolation: 1st order linear
t: 60002-element Array{Float64,1}:
 0.0
 5.0e-5
 0.0001
 0.00015000000000000001
 0.0002
 0.00025
 0.00030000000000000003
 0.00035000000000000005
 0.0004000000000000001
 0.0004500000000000001
 ⋮
 2.9996499999996784
 2.9996999999996783
 2.999749999999678
 2.999799999999678
 2.999849999999678
 2.999899999999678
 2.9999499999996777
 2.9999999999996776
 3.0
u: 60002-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.49224898000189354, 0.9922489800018935, 0.9922489800018935, 0.9922489800
018935, 0.9922489800018935, 0.9922489800018935, 0.9922489800018935, 0.99224
89800018935, 0.9922489800018935, 0.9922489800018935  …  0.9922489800018935,
 0.9922489800018935, 0.9922489800018935, 0.9922489800018935, 0.992248980001
8935, 0.9922489800018935, 0.9922489800018935, 0.9922489800018935, 0.9922489
800018935, 0.49224898000189354]
 [0.4926721630086565, 0.7352899652026758, 0.9852899652026758, 0.98528996520
26758, 0.9852899652026758, 0.9852899652026758, 0.9852899652026758, 0.985289
9652026758, 0.9852899652026758, 0.9852899652026758  …  0.9852899652026758, 
0.9852899652026758, 0.9852899652026758, 0.9852899652026758, 0.9852899652026
758, 0.9852899652026758, 0.9852899652026758, 0.9852899652026758, 0.73528996
52026758, 0.4926721630086565]
 [0.3652767465857273, 0.7354465835560122, 0.8555537544795794, 0.98055375447
95794, 0.9805537544795794, 0.9805537544795794, 0.9805537544795794, 0.980553
7544795794, 0.9805537544795794, 0.9805537544795794  …  0.9805537544795794, 
0.9805537544795794, 0.9805537544795794, 0.9805537544795794, 0.9805537544795
794, 0.9805537544795794, 0.9805537544795794, 0.8555537544795794, 0.73544658
35560122, 0.3652767465857273]
 [0.3657130587256266, 0.6063678562094648, 0.8532917870815846, 0.91265745814
9977, 0.975157458149977, 0.975157458149977, 0.975157458149977, 0.9751574581
49977, 0.975157458149977, 0.975157458149977  …  0.975157458149977, 0.975157
458149977, 0.975157458149977, 0.975157458149977, 0.975157458149977, 0.97515
7458149977, 0.912657458149977, 0.8532917870815846, 0.6063678562094648, 0.36
57130587256266]
 [0.3034722802698041, 0.6099805230971759, 0.7601854483977877, 0.91494422162
31104, 0.9446763362571008, 0.9759263362571008, 0.9759263362571008, 0.975926
3362571008, 0.9759263362571008, 0.9759263362571008  …  0.9759263362571008, 
0.9759263362571008, 0.9759263362571008, 0.9759263362571008, 0.9759263362571
008, 0.9446763362571008, 0.9149442216231104, 0.7601854483977877, 0.60998052
30971759, 0.3034722802698041]
 [0.3064606628001332, 0.5347843768563332, 0.7661456663787495, 0.85686403240
65733, 0.9500124787617917, 0.9650299503663454, 0.9806549503663454, 0.980654
9503663454, 0.9806549503663454, 0.9806549503663454  …  0.9806549503663454, 
0.9806549503663454, 0.9806549503663454, 0.9806549503663454, 0.9650299503663
454, 0.9500124787617917, 0.8568640324065733, 0.7661456663787495, 0.53478437
68563332, 0.3064606628001332]
 [0.26692656968252826, 0.5354906438944077, 0.6946601668691639, 0.8567772025
28268, 0.9095035969401375, 0.9638675034331279, 0.9713524995070033, 0.979164
9995070033, 0.9791649995070033, 0.9791649995070033  …  0.9791649995070033, 
0.9791649995070033, 0.9791649995070033, 0.9713524995070033, 0.9638675034331
279, 0.9095035969401375, 0.856777202528268, 0.694660166869164, 0.5354906438
944077, 0.26692656968252826]
 [0.2680907100022484, 0.4814862632252294, 0.6970327746229348, 0.80319050369
50542, 0.9114991997770914, 0.9416752388750195, 0.9727731272878848, 0.976525
7342629594, 0.9804319842629594, 0.9804319842629594  …  0.9804319842629594, 
0.9804319842629594, 0.9765257342629594, 0.9727731272878848, 0.9416752388750
192, 0.9114991997770914, 0.8031905036950543, 0.6970327746229348, 0.48148626
32252294, 0.2680907100022484]
 [0.24118812022681135, 0.48336093414219855, 0.6434953487590384, 0.805599157
7122333, 0.8739458172791251, 0.9436991970280049, 0.9607151376914994, 0.9782
234356381126, 0.9801062228892022, 0.9820593478892022  …  0.9820593478892022
, 0.9801062228892022, 0.9782234356381126, 0.9607151376914994, 0.94369919702
80046, 0.873945817279125, 0.8055991577122333, 0.6434953487590384, 0.4833609
3414219855, 0.24118812022681135]
 ⋮
 [1.052002068231365e-14, 2.102986392404198e-14, 3.1519362130615924e-14, 4.1
97836738997438e-14, 5.239676129001272e-14, 6.276446470749439e-14, 7.3071447
55894997e-14, 8.330773850413722e-14, 9.346343459266631e-14, 1.0352871084446
77e-13  …  1.0352871084446738e-13, 9.346343459266665e-14, 8.330773850413698
e-14, 7.30714475589502e-14, 6.27644647074942e-14, 5.23967612900129e-14, 4.1
97836738997426e-14, 3.151936213061605e-14, 2.102986392404192e-14, 1.0520020
682313691e-14]
 [1.0623950826408884e-14, 2.1237623666530154e-14, 3.183075047736433e-14, 4.
239308309286795e-14, 5.291440313839328e-14, 6.338453191630452e-14, 7.379334
025321209e-14, 8.41307582992917e-14, 9.438678527021604e-14, 1.0455149912226
483e-13  …  1.0455149912226519e-13, 9.438678527021574e-14, 8.4130758299292e
-14, 7.379334025321189e-14, 6.338453191630473e-14, 5.2914403138393114e-14, 
4.2393083092868104e-14, 3.1830750477364244e-14, 2.1237623666530236e-14, 1.0
623950826408856e-14]
 [1.0502706313124719e-14, 2.099525193619807e-14, 3.14674860089799e-14, 4.19
0927732134356e-14, 5.2310524114566396e-14, 6.266116385413025e-14, 7.2951182
96457208e-14, 8.317062651697348e-14, 9.330960785970904e-14, 1.0335831818314
632e-13  …  1.03358318183146e-13, 9.330960785970938e-14, 8.317062651697322e
-14, 7.295118296457235e-14, 6.266116385413006e-14, 5.2310524114566585e-14, 
4.190927732134344e-14, 3.146748600898002e-14, 2.099525193619801e-14, 1.0502
70631312476e-14]
 [1.0571722572906842e-14, 2.1133217686988224e-14, 3.1674267777824774e-14, 4
.2184675060236283e-14, 5.265427139398125e-14, 6.307292812077452e-14, 7.3430
5658631117e-14, 8.37171642754139e-14, 9.392277173806738e-14, 1.040375149849
6983e-13  …  1.0403751498497022e-13, 9.392277173806709e-14, 8.3717164275414
18e-14, 7.343056586311147e-14, 6.307292812077474e-14, 5.2654271393981095e-1
4, 4.218467506023644e-14, 3.1674267777824685e-14, 2.1133217686988306e-14, 1
.0571722572906812e-14]
 [1.0539677902699127e-14, 2.106915934772175e-14, 3.1578257741805435e-14, 4.
205680621097352e-14, 5.249466743632806e-14, 6.288174346125253e-14, 7.320798
546053099e-14, 8.34634034619395e-14, 9.363807601089664e-14, 1.0372215976883
297e-13  …  1.0372215976883266e-13, 9.363807601089698e-14, 8.34634034619392
4e-14, 7.320798546053123e-14, 6.288174346125235e-14, 5.249466743632825e-14,
 4.205680621097339e-14, 3.157825774180556e-14, 2.106915934772169e-14, 1.053
9677902699168e-14]
 [1.0620812765513377e-14, 2.1231350580610387e-14, 3.18213484352258e-14, 4.2
380561190379183e-14, 5.289877348968623e-14, 6.336580964205489e-14, 7.377154
346601106e-14, 8.410590808612318e-14, 9.435890567205629e-14, 1.045206171108
2456e-13  …  1.0452061711082494e-13, 9.4358905672056e-14, 8.410590808612345
e-14, 7.377154346601085e-14, 6.336580964205509e-14, 5.2898773489686076e-14,
 4.238056119037935e-14, 3.18213484352257e-14, 2.1231350580610472e-14, 1.062
0812765513347e-14]
 [1.0599460064917246e-14, 2.1188665836777017e-14, 3.1757372942887885e-14, 4
.229535684169408e-14, 5.279242271435806e-14, 6.323841532759069e-14, 7.36232
2885818193e-14, 8.393681666973403e-14, 9.416920103213119e-14, 1.04310482774
35158e-13  …  1.0431048277435128e-13, 9.416920103213152e-14, 8.393681666973
38e-14, 7.362322885818216e-14, 6.323841532759051e-14, 5.279242271435824e-14
, 4.229535684169396e-14, 3.175737294288801e-14, 2.118866583677695e-14, 1.05
99460064917288e-14]
 [1.0546493843846312e-14, 2.1082784636033157e-14, 3.15986791956948e-14, 4.2
084004074002815e-14, 5.2528615396321713e-14, 6.292240867575131e-14, 7.32553
2858856669e-14, 8.351737870209277e-14, 9.369863114560929e-14, 1.03789236214
92219e-13  …  1.0378923621492257e-13, 9.369863114560901e-14, 8.351737870209
305e-14, 7.32553285885665e-14, 6.29224086757515e-14, 5.252861539632157e-14,
 4.208400407400296e-14, 3.159867919569471e-14, 2.108278463603324e-14, 1.054
6493843846279e-14]
 [1.0546492486559226e-14, 2.108278192277207e-14, 3.159867512908462e-14, 4.2
08399865797772e-14, 5.252860863612136e-14, 6.292240057791576e-14, 7.3255319
16093007e-14, 8.35173679537757e-14, 9.36986190870101e-14, 1.037892228577067
8e-13  …  1.0378922285770716e-13, 9.369861908700983e-14, 8.351736795377598e
-14, 7.325531916092988e-14, 6.292240057791595e-14, 5.252860863612122e-14, 4
.2083998657977864e-14, 3.159867512908453e-14, 2.1082781922772153e-14, 1.054
6492486559193e-14]
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.1);
19.385573 seconds (54.54 M allocations: 8.460 GiB, 6.26% gc time)
retcode: Success
Interpolation: 1st order linear
t: 68-element Array{Float64,1}:
 0.0
 0.0008
 0.00096
 0.00114
 0.0013425
 0.0015703125
 0.0018266015625
 0.0021149267578125003
 0.002439292602539063
 0.002804204177856446
 ⋮
 1.1854423983416045
 1.333682698134305
 1.5004530354010932
 1.6880696648262299
 1.8991383729295088
 2.1365906695456975
 2.4037245032389096
 2.7042500661437736
 3.0
u: 68-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.2988197149343732, 0.5020663317111074, 0.645043179363209, 0.745622363847
1902, 0.8163762832235489, 0.86614917741433, 0.9011626581933811, 0.925793410
6580838, 0.9431202788665255, 0.9553091213447618  …  0.9553091213447616, 0.9
431202788665255, 0.9257934106580837, 0.9011626581933811, 0.8661491774143301
, 0.8163762832235489, 0.7456223638471902, 0.6450431793632088, 0.50206633171
11074, 0.2988197149343731]
 [0.23706397357240147, 0.4340722094457796, 0.5866616965447955, 0.7004772665
288229, 0.7835362547683794, 0.8433496604680124, 0.8860664126134938, 0.91641
21443908664, 0.9378961718146078, 0.9530727204433849  …  0.9530727204433848,
 0.9378961718146079, 0.9164121443908663, 0.8860664126134938, 0.843349660468
0123, 0.7835362547683794, 0.7004772665288229, 0.5866616965447955, 0.4340722
094457795, 0.23706397357240144]
 [0.19830970602563924, 0.3767684019953319, 0.525865545284988, 0.64430983336
50287, 0.735178522319983, 0.8032024317687791, 0.8532432073601421, 0.8895964
520274147, 0.9157685481813513, 0.9344882987246346  …  0.9344882987246347, 0
.9157685481813514, 0.8895964520274146, 0.8532432073601421, 0.80320243176877
91, 0.7351785223199832, 0.6443098333650287, 0.5258655452849879, 0.376768401
9953319, 0.19830970602563924]
 [0.17329052947149123, 0.3346538980146704, 0.4760836380236111, 0.5941976150
207467, 0.6891786593888578, 0.7633373036605012, 0.8199204432593554, 0.86232
31498604549, 0.8936544861634113, 0.916551059340171  …  0.9165510593401712, 
0.8936544861634115, 0.8623231498604549, 0.8199204432593555, 0.7633373036605
012, 0.6891786593888578, 0.5941976150207467, 0.47608363802361114, 0.3346538
980146704, 0.1732905294714912]
 [0.1568922640857151, 0.3058025425498876, 0.44048899343600845, 0.5572770489
138719, 0.6549708513740654, 0.7342628738972161, 0.7970194090885682, 0.84565
98088068409, 0.8827097822041591, 0.9105274156330955  …  0.9105274156330955,
 0.8827097822041593, 0.845659808806841, 0.7970194090885683, 0.7342628738972
161, 0.6549708513740654, 0.5572770489138719, 0.44048899343600856, 0.3058025
4254988765, 0.15689226408571508]
 [0.13922344106332848, 0.2729109165271084, 0.39640326017757205, 0.506478360
0898846, 0.6014976472521902, 0.6812271556290448, 0.7464863643665689, 0.7987
592546816052, 0.8398523486909415, 0.8716368849234541  …  0.8716368849234544
, 0.8398523486909417, 0.7987592546816054, 0.7464863643665691, 0.68122715562
90451, 0.6014976472521902, 0.5064783600898846, 0.39640326017757205, 0.27291
09165271084, 0.13922344106332848]
 [0.12700314834636578, 0.24992426042535737, 0.3651759374457938, 0.470031401
328785, 0.5627946656729537, 0.6427815141667893, 0.7101652131334445, 0.76575
43254435694, 0.8107601033905357, 0.846591350935032  …  0.8465913509350322, 
0.8107601033905358, 0.7657543254435695, 0.7101652131334446, 0.6427815141667
895, 0.5627946656729538, 0.4700314013287851, 0.3651759374457939, 0.24992426
04253574, 0.1270031483463658]
 [0.11401500257758088, 0.22500478860836132, 0.33023577418808525, 0.42749971
020140376, 0.515251907181259, 0.5926453555123323, 0.6594765795418202, 0.716
0724433673782, 0.7631494213421809, 0.8016716385526138  …  0.801671638552613
9, 0.763149421342181, 0.7160724433673783, 0.6594765795418204, 0.59264535551
23324, 0.5152519071812589, 0.4274997102014038, 0.33023577418808525, 0.22500
478860836134, 0.11401500257758085]
 [0.10524708449543878, 0.20814913577381083, 0.30654564439849746, 0.39861621
154578786, 0.4829859978289853, 0.5587720134644749, 0.6255734017225407, 0.68
34176829602422, 0.7326790893986714, 0.7739850910098597  …  0.77398509100985
98, 0.7326790893986715, 0.6834176829602422, 0.6255734017225408, 0.558772013
4644749, 0.4829859978289854, 0.39861621154578786, 0.30654564439849746, 0.20
814913577381086, 0.1052470844954388]
 ⋮
 [9.332995319509613e-7, 1.8656961568828872e-6, 2.796287841272227e-6, 3.7241
742977795787e-6, 4.648457856161631e-6, 5.56824433180409e-6, 6.4826438906465
44e-6, 7.390771910238935e-6, 8.291749835280061e-6, 9.1847060279392e-6  …  9
.184706027924173e-6, 8.291749835296575e-6, 7.390771910222025e-6, 6.48264389
0661985e-6, 5.568244331791514e-6, 4.648457856170242e-6, 3.724174297774798e-
6, 2.7962878412736328e-6, 1.8656961568834708e-6, 9.332995319500348e-7]
 [5.098976287287976e-7, 1.0193019644325762e-6, 1.5277201913174505e-6, 2.034
660447581408e-6, 2.5396323012720208e-6, 3.042147224726071e-6, 3.54171906722
0616e-6, 4.037864525246996e-6, 4.530103610138736e-6, 5.017960112343954e-6  
…  5.017960112347383e-6, 4.530103610134983e-6, 4.037864525250852e-6, 3.5417
190672171065e-6, 3.042147224728939e-6, 2.539632301270064e-6, 2.034660447582
499e-6, 1.527720191317132e-6, 1.0193019644324445e-6, 5.098976287290088e-7]
 [-5.15214774822921e-9, -1.0299311111314894e-8, -1.5436510591535653e-8, -2.
0558776181143158e-8, -2.5661152581379992e-8, -3.073870333142406e-8, -3.5786
516570079054e-8, -4.079970839096552e-8, -4.577342949809753e-8, -5.070286730
132998e-8  …  -5.070286739902655e-8, -4.5773429386938325e-8, -4.07997085035
0401e-8, -3.5786516466773857e-8, -3.07387034156167e-8, -2.5661152522216064e
-8, -2.05587762146278e-8, -1.5436510579563637e-8, -1.0299311110111128e-8, -
5.1521477524136985e-9]
 [1.0688248007243731e-9, 2.136615574195394e-9, 3.202339329758157e-9, 4.2649
64993585356e-9, 5.323464634032697e-9, 6.376814087300223e-9, 7.4239944950623
89e-9, 8.463992515540273e-9, 9.495802374600166e-9, 1.0518425399097564e-8  …
  1.0518425453278007e-8, 9.495802312952562e-9, 8.463992577952195e-9, 7.4239
94437770657e-9, 6.376814133992096e-9, 5.32346460122092e-9, 4.26496501215555
9e-9, 3.202339323118239e-9, 2.136615573528122e-9, 1.0688248030447368e-9]
 [1.8393458663282709e-10, 3.6769509540193704e-10, 5.510910403751985e-10, 7.
339693085070227e-10, 9.161136292802932e-10, 1.0974061308973848e-9, 1.277588
780153546e-9, 1.4566015030087716e-9, 1.6341175144855598e-9, 1.8101609333594
473e-9  …  1.8101389454916057e-9, 1.6341431083602111e-9, 1.456575495625197e
-9, 1.277612645612092e-9, 1.0973865544041782e-9, 9.1612759367238e-10, 7.339
611367325401e-10, 5.510943673950017e-10, 3.676947768281174e-10, 1.839340203
8289335e-10]
 [7.016379380376585e-11, 1.4026056822131378e-10, 2.1021968405491675e-10, 2.
799788642419346e-10, 3.494618749317331e-10, 4.1861446410010215e-10, 4.87351
3605358491e-10, 5.55631459725158e-10, 6.233545617268015e-10, 6.904986844324
171e-10  …  6.90493796786724e-10, 6.233602509121874e-10, 5.556256786276051e
-10, 4.8735666551227e-10, 4.1861011251123735e-10, 3.494649790149981e-10, 2.
799770477890282e-10, 2.1022042357574352e-10, 1.4026049744032841e-10, 7.0163
66791122048e-11]
 [1.6891473959400596e-11, 3.332832977170069e-11, 5.071174162952577e-11, 6.6
36299737583262e-11, 8.453289250291253e-11, 9.889341935336633e-11, 1.1836382
866420515e-10, 1.3063810381794496e-10, 1.5202540628407604e-10, 1.6205717866
987973e-10  …  1.642679679390813e-10, 1.4940307522993188e-10, 1.33321012560
65718e-10, 1.1589664590635423e-10, 1.009287973666756e-10, 8.306082329702845
e-11, 6.725229021128392e-11, 5.0311325898080035e-11, 3.3418176490510484e-11
, 1.6912988446426044e-11]
 [6.120338851097766e-12, 1.2125847876905982e-11, 1.836285165815406e-11, 2.4
163850263091033e-11, 3.058329471886448e-11, 3.6046865277000544e-11, 4.27691
1687247028e-11, 4.7690242360217294e-11, 5.486118527880222e-11, 5.9193791440
11628e-11  …  5.974316733147388e-11, 5.420954414867202e-11, 4.8356936159307
4e-11, 4.2156030455797076e-11, 3.6552649261692355e-11, 3.021749167543232e-1
1, 2.4384834774834616e-11, 1.8263351879035143e-11, 1.2148172309692605e-11, 
6.1256865370865235e-12]
 [5.207362542717675e-12, -2.3877004723904866e-12, 1.8153870660609373e-11, -
9.105414844411234e-12, 3.659190973205542e-11, -2.2884648696383787e-11, 6.40
2183312715572e-11, -4.644789117001646e-11, 9.672155960505225e-11, -6.267551
366088518e-11  …  -4.5829401842594084e-12, 2.6800383602476348e-11, 2.566015
50741217e-11, -2.5506903182869807e-12, 3.23593226358423e-11, -3.88522884093
7231e-12, 1.6077005921273914e-11, 5.875665633536463e-12, 1.466840668937802e
-12, 4.883192311934301e-12]
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.01);
0.007071 seconds (7.13 k allocations: 875.234 KiB)
retcode: Success
Interpolation: 1st order linear
t: 80-element Array{Float64,1}:
 0.0
 0.00020599494558788735
 0.0002471939347054648
 0.0002935427974627395
 0.00034568526806467347
 0.00040434554749184924
 0.000470338361847422
 0.0005445802779974413
 0.000628102433666213
 0.0007220648587935812
 ⋮
 1.2549035454019108
 1.411781938198069
 1.5882701300937467
 1.7868193459763841
 2.0101872138443513
 2.2614760651958146
 2.5441760229662105
 2.862213475457906
 3.0
u: 80-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.4975842225537368, 0.7616401451434134, 0.894965721958418, 0.962283694828
4142, 0.9962734931820604, 1.0134354239839969, 1.0221007260192139, 1.0264759
597984492, 1.0286850773277822, 1.029800492070091  …  1.0298004920700912, 1.
0286850773277822, 1.0264759597984492, 1.0221007260192139, 1.013435423983996
9, 0.9962734931820605, 0.9622836948284142, 0.8949657219584181, 0.7616401451
434135, 0.49758422255373685]
 [0.4369643991463944, 0.7168810998519246, 0.8723780374292617, 0.95427205714
34514, 0.996428817826598, 1.0179071082283344, 1.028797819014686, 1.03430767
98795867, 1.0370923097617648, 1.0384989350993639  …  1.038498935099364, 1.0
370923097617648, 1.0343076798795867, 1.0287978190146863, 1.0179071082283344
, 0.996428817826598, 0.9542720571434515, 0.8723780374292618, 0.716881099851
9246, 0.43696439914639446]
 [0.39004835794427806, 0.6721839315954988, 0.844898901767729, 0.94183833399
50821, 0.9937412135841441, 1.020812911714539, 1.0347283301936807, 1.0418232
887150598, 1.0454245755161005, 1.04724806186531  …  1.0472480618653102, 1.0
454245755161005, 1.0418232887150598, 1.034728330193681, 1.020812911714539, 
0.9937412135841441, 0.9418383339950822, 0.8448989017677291, 0.6721839315954
989, 0.39004835794427806]
 [0.3503618104414706, 0.6244358214253071, 0.8069100704816259, 0.91671678847
61424, 0.9786969393382601, 1.01227815016828, 1.03000139667984, 1.0391998643
991471, 1.0439235820727604, 1.0463333272349762  …  1.0463333272349764, 1.04
39235820727604, 1.039199864399147, 1.03000139667984, 1.01227815016828, 0.97
86969393382602, 0.9167167884761425, 0.806910070481626, 0.6244358214253071, 
0.35036181044147063]
 [0.3188656329724629, 0.5822982246453482, 0.7703853462998016, 0.89141741651
22655, 0.9638002274424555, 1.0049094551624826, 1.027421471447381, 1.0394372
341143094, 1.0457366564277875, 1.048998440122505  …  1.0489984401225052, 1.
0457366564277875, 1.0394372341143094, 1.027421471447381, 1.0049094551624824
, 0.9638002274424555, 0.8914174165122656, 0.7703853462998016, 0.58229822464
53482, 0.31886563297246295]
 [0.2892401262306874, 0.5374981116651163, 0.7250105319865396, 0.85321399728
38871, 0.9344606431948427, 0.983070387140852, 1.0109135260125885, 1.0263447
79696487, 1.0346872296127758, 1.0391140594838433  …  1.0391140594838435, 1.
0346872296127758, 1.026344779696487, 1.0109135260125885, 0.9830703871408519
, 0.9344606431948427, 0.8532139972838871, 0.7250105319865396, 0.53749811166
51163, 0.2892401262306874]
 [0.2677015279045413, 0.5040615582201857, 0.6909092628446128, 0.82562677141
52957, 0.9157808368824185, 0.9726144169194552, 1.0067740460887828, 1.026539
639392467, 1.0376355789385925, 1.0437167098252464  …  1.0437167098252467, 1
.0376355789385925, 1.0265396393924668, 1.0067740460887828, 0.97261441691945
52, 0.9157808368824185, 0.8256267714152956, 0.6909092628446128, 0.504061558
2201857, 0.2677015279045413]
 [0.2470087981383067, 0.46972879113358357, 0.65229986390069, 0.790025281606
8292, 0.8868531314135825, 0.9510329109374928, 0.9915389999783059, 1.0160856
634683288, 1.0304682594988237, 1.0386636711384418  …  1.038663671138442, 1.
0304682594988237, 1.0160856634683288, 0.9915389999783057, 0.951032910937492
8, 0.8868531314135825, 0.7900252816068292, 0.6522998639006901, 0.4697287911
3358357, 0.2470087981383067]
 [0.23064304953712567, 0.44208418250790843, 0.6206622066781381, 0.760716357
1584481, 0.8636303773006149, 0.935104886851013, 0.9823951932173796, 1.01241
20762513835, 1.0308006583118399, 1.041729275161687  …  1.0417292751616871, 
1.0308006583118399, 1.0124120762513835, 0.9823951932173796, 0.9351048868510
129, 0.8636303773006149, 0.7607163571584481, 0.6206622066781382, 0.44208418
25079084, 0.23064304953712567]
 ⋮
 [4.367989621211587e-6, 8.731753494570772e-6, 1.3087069960341152e-5, 1.7429
725531172798e-5, 2.1755518968181503e-5, 2.6060265345706505e-5, 3.0339800099
531213e-5, 3.458998305627061e-5, 3.880670243831924e-5, 4.298587884208621e-5
  …  4.29858788420825e-5, 3.8806702438305894e-5, 3.458998305628753e-5, 3.03
3980009951962e-5, 2.6060265345704784e-5, 2.1755518968194164e-5, 1.742972553
1158056e-5, 1.308706996035069e-5, 8.731753494566137e-6, 4.3679896212133505e
-6]
 [1.8584691451531362e-6, 3.715140341436371e-6, 5.568217379375286e-6, 7.4159
07526623111e-6, 9.256423262278402e-6, 1.1087984006254724e-5, 1.290881784180
153e-5, 1.4717163229797346e-5, 1.6511270712853716e-5, 1.828940460786535e-5 
 …  1.8289404607864676e-5, 1.651127071285133e-5, 1.4717163229800334e-5, 1.2
908817841799457e-5, 1.1087984006254409e-5, 9.256423262280641e-6, 7.41590752
6620489e-6, 5.568217379376975e-6, 3.715140341435547e-6, 1.8584691451534488e
-6]
 [9.571013028671739e-7, 1.913276672012931e-6, 2.867601069617109e-6, 3.81915
1248242107e-6, 4.7670066449077925e-6, 5.710250270268037e-6, 6.6479695975171
73e-6, 7.579257443326324e-6, 8.50321284724114e-6, 9.418941941734525e-6  …  
9.418941941763057e-6, 8.50321284728497e-6, 7.579257443261756e-6, 6.64796959
7559415e-6, 5.710250270272952e-6, 4.767006644861907e-6, 3.819151248298852e-
6, 2.8676010695748955e-6, 1.9132766720355043e-6, 9.571013028582093e-7]
 [4.793310783063639e-7, 9.581984347505866e-7, 1.4361387960963608e-6, 1.9126
89785902555e-6, 2.387390371878839e-6, 2.85978131184738e-6, 3.32940559810229
8e-6, 3.7958088994589016e-6, 4.258540000861171e-6, 4.717151239839426e-6  … 
 4.717151239840602e-6, 4.258540000862972e-6, 3.7958088994562902e-6, 3.32940
55981040292e-6, 2.8597813118475918e-6, 2.3873903718769816e-6, 1.91268978590
487e-6, 1.4361387960946502e-6, 9.58198434751507e-7, 4.793310783060012e-7]
 [2.2290128665624526e-7, 4.455869307428469e-7, 6.678414981753724e-7, 8.8944
99722317214e-7, 1.1101979606819625e-6, 1.3298719045378637e-6, 1.54825928298
88986e-6, 1.7651488207793117e-6, 1.9803306909061724e-6, 2.193596719154023e-
6  …  2.19359671911762e-6, 1.9803306908760536e-6, 1.765148820831766e-6, 1.5
482592829537785e-6, 1.3298719045345806e-6, 1.1101979607193616e-6, 8.8944997
21825728e-7, 6.678414982153145e-7, 4.455869307193434e-7, 2.2290128666613193
e-7]
 [9.381894310882206e-8, 1.8754712245670328e-7, 2.810938620571076e-7, 3.7436
86615354001e-7, 4.672812834614588e-7, 5.597418409824054e-7, 6.5166088436629
31e-7, 7.429494879820063e-7, 8.335193359059027e-7, 9.232828077458956e-7  … 
 9.232828077368218e-7, 8.335193358983946e-7, 7.429494879950879e-7, 6.516608
843575377e-7, 5.597418409815882e-7, 4.672812834707857e-7, 3.743686615231459
7e-7, 2.8109386206706777e-7, 1.875471224508435e-7, 9.38189431112874e-8]
 [3.942256256947025e-8, 7.880698611546716e-8, 1.1811516956721927e-7, 1.5730
908322315219e-7, 1.9635081188173717e-7, 2.352025822280736e-7, 2.73826810840
27435e-7, 3.121861282561845e-7, 3.502434270513409e-7, 3.8796188670311543e-7
  …  3.8796188700357327e-7, 3.502434271862142e-7, 3.1218612795446417e-7, 2.
7382681105213666e-7, 2.3520258224372794e-7, 1.9635081165600374e-7, 1.573090
8353493342e-7, 1.1811516929627799e-7, 7.880698628758788e-8, 3.9422562492510
64e-8]
 [1.734915951967894e-8, 3.4681534783064076e-8, 5.198035805687418e-8, 6.9228
89342171849e-8, 8.64104546765339e-8, 1.0350841896934422e-7, 1.2050624597718
421e-7, 1.373874905851572e-7, 1.5413582201934413e-7, 1.7073503666316152e-7 
 …  1.707350367443707e-7, 1.5413582205580397e-7, 1.3738749050360232e-7, 1.2
050624603445116e-7, 1.0350841897357648e-7, 8.641045461551893e-8, 6.92288935
0599215e-8, 5.198035798364122e-8, 3.4681534829586434e-8, 1.7349159498878027
e-8]
 [8.216973070961227e-9, 1.6425996739502e-8, 2.461912934496659e-8, 3.2788444
48437282e-8, 4.092603898837761e-8, 4.902404011600175e-8, 5.707461372736773e
-8, 6.506997126609e-8, 7.300237787299457e-8, 8.086415933353167e-8  …  8.086
415934811274e-8, 7.300237787954431e-8, 6.506997125144358e-8, 5.707461373765
218e-8, 4.902404011676208e-8, 4.092603897742034e-8, 3.278844449950659e-8, 2
.4619129331816278e-8, 1.6425996747855566e-8, 8.216973067226393e-9]
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.001);
0.045407 seconds (6.32 k allocations: 808.922 KiB, 87.03% gc time)
retcode: Success
Interpolation: 1st order linear
t: 64-element Array{Float64,1}:
 0.0
 0.0002434199984833298
 0.0005172674967770758
 0.0008253459323575401
 0.0011719341723855625
 0.0015618459424170878
 0.002000496683702554
 0.002493978767648703
 0.003049146112088121
 0.003673709374582466
 ⋮
 1.265222915592534
 1.4236192000400842
 1.6018150200435781
 1.8022853175475089
 2.027814402239431
 2.2815346225178432
 2.5669698703310573
 2.8880845241209228
 3.0
u: 64-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.4726693753722989, 0.7202203693061271, 0.8520042218963426, 0.92215939892
69059, 0.9595065303656409, 0.9793882880026363, 0.9899723481836613, 0.995606
7760828884, 0.9986062656179634, 1.0002030448530403  …  1.0002030448530406, 
0.9986062656179635, 0.9956067760828884, 0.9899723481836614, 0.9793882880026
364, 0.959506530365641, 0.9221593989269059, 0.8520042218963427, 0.720220369
3061273, 0.4726693753722989]
 [0.29374527348664947, 0.5252905162133772, 0.6899252135422879, 0.8003758660
636972, 0.8717441832900632, 0.9166632103995931, 0.944391239687183, 0.961253
490284521, 0.9713869992649493, 0.9774183487796863  …  0.9774183487796864, 0
.9713869992649495, 0.9612534902845211, 0.9443912396871834, 0.91666321039959
34, 0.8717441832900632, 0.8003758660636973, 0.6899252135422879, 0.525290516
2133773, 0.29374527348664947]
 [0.21378851452257117, 0.4045482931370362, 0.5609478225491468, 0.6815256190
948351, 0.7703227406983193, 0.8334577293207311, 0.8771219365748565, 0.90665
46134570143, 0.9262671216813422, 0.9390942885020827  …  0.9390942885020828,
 0.9262671216813423, 0.9066546134570143, 0.8771219365748564, 0.833457729320
731, 0.7703227406983194, 0.6815256190948351, 0.5609478225491469, 0.40454829
31370364, 0.21378851452257125]
 [0.17094579628209866, 0.33065437912912915, 0.47115473563822996, 0.58866215
6721113, 0.6829061261417441, 0.7559101409869612, 0.8108429540267954, 0.8511
78596527968, 0.8801854473851076, 0.9006754137649804  …  0.9006754137649804,
 0.8801854473851078, 0.8511785965279682, 0.8108429540267954, 0.755910140986
9612, 0.6829061261417441, 0.588662156721113, 0.47115473563822985, 0.3306543
7912912915, 0.17094579628209866]
 [0.14636411428398302, 0.2861023706497021, 0.41381596086997696, 0.526011447
9282864, 0.6211741121915191, 0.6994379719485393, 0.7620880422078743, 0.8110
654458646552, 0.8485651393460881, 0.8767541371999612  …  0.8767541371999612
, 0.8485651393460882, 0.8110654458646553, 0.7620880422078744, 0.69943797194
85393, 0.621174112191519, 0.5260114479282864, 0.4138159608699769, 0.2861023
706497021, 0.14636411428398302]
 [0.12896244464497317, 0.2536473885840704, 0.37031695190970404, 0.476162440
0448438, 0.5694727138889033, 0.649598353731563, 0.7167769765916662, 0.77189
37963902325, 0.8162370480832744, 0.8512857304225832  …  0.8512857304225829,
 0.8162370480832742, 0.7718937963902321, 0.7167769765916658, 0.649598353731
5627, 0.5694727138889032, 0.4761624400448437, 0.370316951909704, 0.25364738
85840703, 0.12896244464497314]
 [0.11749233043600558, 0.231986089658872, 0.3407623855074899, 0.44160680205
187636, 0.53294416009128, 0.6138754450328207, 0.6841315361795242, 0.7439703
548041618, 0.7940461209908012, 0.8352749287761193  …  0.8352749287761192, 0
.794046120990801, 0.7439703548041616, 0.684131536179524, 0.6138754450328207
, 0.53294416009128, 0.44160680205187636, 0.34076238550748994, 0.23198608965
887207, 0.11749233043600561]
 [0.10474680629601679, 0.20737028320087472, 0.30589961095804086, 0.39864689
48682841, 0.48429895479419105, 0.5619628099678816, 0.6311661987287565, 0.69
18212588933924, 0.74416305784284, 0.7886752125866168  …  0.7886752125866165
, 0.7441630578428398, 0.6918212588933922, 0.6311661987287563, 0.56196280996
78815, 0.484298954794191, 0.3986468948682841, 0.30589961095804086, 0.207370
28320087475, 0.1047468062960168]
 [0.09984766949564494, 0.19807004058208738, 0.29313333813625864, 0.38367675
827986936, 0.46857562151080784, 0.5469811888434081, 0.6183357594546073, 0.6
823649885771697, 0.7390517758123326, 0.7885973500122875  …  0.7885973500122
871, 0.7390517758123323, 0.6823649885771694, 0.6183357594546073, 0.54698118
8843408, 0.4685756215108079, 0.3836767582798694, 0.2931333381362586, 0.1980
7004058208735, 0.09984766949564491]
 ⋮
 [7.6006893707978865e-6, 1.5194025565512253e-5, 2.2772662521782618e-5, 3.03
29268397812175e-5, 3.785653266545165e-5, 4.534717318266272e-5, 5.2793943238
5202e-5, 6.018963856393741e-5, 6.752710430132985e-5, 7.479924192648002e-5  
…  7.479924192648019e-5, 6.752710430133037e-5, 6.018963856393764e-5, 5.2793
943238520514e-5, 4.534717318266295e-5, 3.785653266545183e-5, 3.032926839781
2365e-5, 2.27726625217827e-5, 1.5194025565512347e-5, 7.6006893707979255e-6]
 [3.4097032083854647e-6, 6.816107749129598e-6, 1.0215918145836276e-5, 1.360
5845301523584e-5, 1.6982609680599952e-5, 2.0342944481613852e-5, 2.368359879
7674026e-5, 2.700134076147092e-5, 3.029296067194568e-5, 3.355527409939911e-
5  …  3.355527409940499e-5, 3.0292960671940348e-5, 2.700134076147393e-5, 2.
3683598797672684e-5, 2.0342944481614123e-5, 1.698260968060042e-5, 1.3605845
301522629e-5, 1.0215918145837319e-5, 6.816107749129028e-6, 3.40970320838560
45e-6]
 [1.7675247462213816e-6, 3.5333395264045515e-6, 5.295736028793087e-6, 7.053
00924859339e-6, 8.8034591374571e-6, 1.0545392248167745e-5, 1.22771233729415
26e-5, 1.3996977173757654e-5, 1.5703289803138296e-5, 1.7394410513815625e-5 
 …  1.7394410513815398e-5, 1.5703289803138408e-5, 1.3996977173757525e-5, 1.
227712337294153e-5, 1.0545392248167705e-5, 8.803459137457058e-6, 7.05300924
8593397e-6, 5.295736028793038e-6, 3.5333395264045566e-6, 1.767524746221371e
-6]
 [6.424879956020843e-7, 1.2843544255633134e-6, 1.924978325565559e-6, 2.5637
399333617598e-6, 3.2000212883483194e-6, 3.833206829407705e-6, 4.46268399042
6166e-6, 5.087843792896822e-6, 5.7080814350976436e-6, 6.322796877153861e-6 
 …  6.322796877157358e-6, 5.708081435094401e-6, 5.087843792898938e-6, 4.462
683990425192e-6, 3.833206829407954e-6, 3.200021288348626e-6, 2.563739933361
1812e-6, 1.924978325566186e-6, 1.2843544255629306e-6, 6.424879956022345e-7]
 [3.012139802730063e-7, 6.021365554737064e-7, 9.024766024453346e-7, 1.20194
35615896542e-6, 1.5002477179644137e-6, 1.7971004815639855e-6, 2.09221466651
16093e-6, 2.3853047688928017e-6, 2.6760872429627213e-6, 2.9642807754572614e
-6  …  2.964280775457368e-6, 2.676087242962658e-6, 2.3853047688928703e-6, 2
.092214666511599e-6, 1.7971004815640042e-6, 1.5002477179644315e-6, 1.201943
5615896474e-6, 9.024766024453562e-7, 6.021365554737009e-7, 3.01213980273012
13e-7]
 [1.2535004662888623e-7, 2.5057882518295767e-7, 3.7556518490762436e-7, 5.00
1882095707329e-7, 6.243273344455022e-7, 7.478624629463489e-7, 8.70674082812
3793e-7, 9.92643381738017e-7, 1.1136523622966418e-6, 1.233583956122864e-6  
…  1.2335839561208904e-6, 1.1136523622986153e-6, 9.92643381736649e-7, 8.706
740828130841e-7, 7.478624629461625e-7, 6.24327334445351e-7, 5.0018820957107
96e-7, 3.7556518490726543e-7, 2.5057882518320966e-7, 1.25350046628775e-7]
 [4.338353254538814e-8, 8.672509432485003e-8, 1.2998275517673675e-7, 1.7311
466610764447e-7, 2.1607909977958967e-7, 2.588344908778848e-7, 3.01339476322
3955e-7, 3.4355293528597964e-7, 3.854340289717695e-7, 4.269422401283071e-7 
 …  4.269422401278554e-7, 3.854340289722167e-7, 3.4355293528566613e-7, 3.01
3394763225543e-7, 2.5883449087784094e-7, 2.16079099779554e-7, 1.73114666107
7225e-7, 1.2998275517665432e-7, 8.672509432490694e-8, 4.338353254536257e-8]
 [1.333180901855456e-8, 2.6650720376378185e-8, 3.9943848877267796e-8, 5.319
8334290404735e-8, 6.640135374913573e-8, 7.954013417846105e-8, 9.26019646866
2386e-8, 1.0557420873920012e-7, 1.1844431660754034e-7, 1.3119983720561286e-
7  …  1.3119983722608448e-7, 1.1844431658633752e-7, 1.0557420875495033e-7, 
9.260196467800488e-8, 7.954013418115196e-8, 6.640135375048643e-8, 5.3198334
28690117e-8, 3.994384888109821e-8, 2.6650720373544873e-8, 1.333180901993443
e-8]
 [1.4361395898345626e-9, 2.870889806346718e-9, 4.3028626242008296e-9, 5.730
672698509526e-9, 7.152938717603161e-9, 8.568284732838624e-9, 9.975341479609
395e-9, 1.1372747738137247e-8, 1.2759151584700627e-8, 1.4133211788943536e-8
  …  1.4133211782927164e-8, 1.2759151590932349e-8, 1.137274773350811e-8, 9.
975341482142856e-9, 8.568284732047694e-9, 7.1529387172063096e-9, 5.73067269
9539243e-9, 4.3028626230751e-9, 2.870889807179487e-9, 1.4361395894289881e-9
]
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitEM(),progress=true,dt=0.001);
11.129955 seconds (28.47 M allocations: 7.217 GiB, 7.76% gc time)
retcode: Success
Interpolation: 1st order linear
t: 66-element Array{Float64,1}:
 0.0
 0.001
 0.0012000000000000001
 0.001425
 0.0016781250000000001
 0.0019628906250000002
 0.0022832519531250004
 0.0026436584472656257
 0.003049115753173829
 0.003505255222320558
 ⋮
 1.1706813810781274
 1.3170915537128933
 1.481802997927005
 1.6671033726678808
 1.875566294251366
 2.1100870810327867
 2.3739229661618855
 2.6707383369321214
 3.0
u: 66-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.26573643298760186, 0.4596825178829871, 0.601232863175694, 0.70454250339
49937, 0.7799424025628158, 0.8349725505959426, 0.8751359622976869, 0.904448
9788382231, 0.9258429018716049, 0.9414571237011701  …  0.9414571237011707, 
0.9258429018716051, 0.9044489788382234, 0.8751359622976871, 0.8349725505959
427, 0.7799424025628159, 0.7045425033949937, 0.6012328631756942, 0.45968251
78829872, 0.26573643298760197]
 [0.20635758146942335, 0.38484077168845726, 0.5290428149803392, 0.641256378
3249245, 0.7266390493820053, 0.7906972109199563, 0.838320736923906, 0.87351
4008909916, 0.8994173685344211, 0.918431658988788  …  0.9184316589887882, 0
.8994173685344213, 0.8735140089099165, 0.8383207369239062, 0.79069721091995
65, 0.7266390493820055, 0.6412563783249248, 0.5290428149803394, 0.384840771
68845743, 0.20635758146942346]
 [0.16950804930391303, 0.3256840855869796, 0.46077962777422404, 0.572457850
7789816, 0.6618090099555125, 0.7316158946932785, 0.7852080814702569, 0.8258
226699983129, 0.8563072338623291, 0.8790246017717224  …  0.8790246017717227
, 0.8563072338623293, 0.8258226699983131, 0.7852080814702571, 0.73161589469
32789, 0.6618090099555128, 0.5724578507789817, 0.4607796277742241, 0.325684
08558697964, 0.16950804930391306]
 [0.15013913430197137, 0.2921566640691692, 0.4200263592384827, 0.5305192298
38417, 0.6228575408638203, 0.6979738198024661, 0.7577721781777628, 0.804555
9390228659, 0.8406497678174694, 0.868185201205925  …  0.8681852012059256, 0
.8406497678174699, 0.804555939022866, 0.757772178177763, 0.6979738198024664
, 0.6228575408638205, 0.5305192298384172, 0.42002635923848275, 0.2921566640
691693, 0.1501391343019714]
 [0.131712115651628, 0.25813947244736757, 0.3748985990461592, 0.47903866516
407506, 0.5691284000428818, 0.6450339941119067, 0.7075605930621734, 0.75808
52368276348, 0.7982502687523138, 0.8297407416762437  …  0.829740741676244, 
0.7982502687523141, 0.7580852368276351, 0.7075605930621738, 0.6450339941119
07, 0.5691284000428819, 0.4790386651640753, 0.37489859904615924, 0.25813947
24473677, 0.13171211565162805]
 [0.1189013649718024, 0.23405241376977354, 0.3421718308936425, 0.4407830655
5984853, 0.5283568089466882, 0.6042776911836146, 0.6686937110031876, 0.7223
126281681853, 0.7661956476557654, 0.8015791176909454  …  0.8015791176909457
, 0.7661956476557656, 0.7223126281681854, 0.668693711003188, 0.604277691183
6149, 0.5283568089466883, 0.44078306555984875, 0.3421718308936425, 0.234052
4137697736, 0.11890136497180244]
 [0.10809829436539457, 0.21344099390414573, 0.31353617077486645, 0.40636541
2789105, 0.49050609979666043, 0.565159959104545, 0.6301038370296571, 0.6855
896011025141, 0.7322209996902507, 0.7708300111924528  …  0.7708300111924531
, 0.7322209996902508, 0.6855896011025143, 0.6301038370296572, 0.56515995910
45452, 0.49050609979666054, 0.40636541278910504, 0.3135361707748665, 0.2134
4099390414575, 0.10809829436539457]
 [0.10055981252149504, 0.19899595195712086, 0.2933473397502141, 0.381952513
4329714, 0.4635426001858407, 0.5372827161915713, 0.6027648896997077, 0.6599
630242610907, 0.7091637269636898, 0.7508866134368807  …  0.750886613436881,
 0.7091637269636901, 0.659963024261091, 0.6027648896997079, 0.5372827161915
715, 0.46354260018584076, 0.38195251343297143, 0.29334733975021415, 0.19899
595195712086, 0.10055981252149504]
 [0.09169214020256668, 0.1817599415111163, 0.2686804538446676, 0.3511209346
066896, 0.42800517086171846, 0.49855178705142217, 0.5622838764654351, 0.619
013408095872, 0.6688065277286583, 0.7119369389969514  …  0.7119369389969518
, 0.6688065277286586, 0.6190134080958722, 0.5622838764654353, 0.49855178705
14223, 0.4280051708617186, 0.35112093460668964, 0.26868045384466765, 0.1817
5994151111632, 0.09169214020256668]
 ⋮
 [2.7958704022215882e-6, 5.58903598039744e-6, 8.376794527224438e-6, 1.11564
49066353035e-5, 1.3925310461537774e-5, 1.6680700018202902e-5, 1.94199520749
0631e-5, 2.214041658219508e-5, 2.4839461666356562e-5, 2.751447617558714e-5 
 …  2.7514476175587338e-5, 2.4839461666356745e-5, 2.214041658219508e-5, 1.9
419952074906477e-5, 1.6680700018203003e-5, 1.392531046153786e-5, 1.11564490
6635309e-5, 8.376794527224488e-6, 5.589035980397474e-6, 2.7958704022216056e
-6]
 [4.2496002612146823e-7, 8.495089308632721e-7, 1.2732359905791302e-6, 1.695
7312767047463e-6, 2.1165860523372186e-6, 2.535393167661598e-6, 2.9517474538
420364e-6, 3.3652461149964696e-6, 3.7754891178756194e-6, 4.182079578869323e
-6  …  4.182079578869336e-6, 3.7754891178756312e-6, 3.3652461149964807e-6, 
2.9517474538420457e-6, 2.5353931676616066e-6, 2.1165860523372263e-6, 1.6957
312767047522e-6, 1.273235990579135e-6, 8.495089308632753e-7, 4.249600261214
698e-7]
 [1.2182515791894566e-7, 2.435324578655562e-7, 3.6500415588748145e-7, 4.861
227359619987e-7, 6.067710236852152e-7, 7.268322996306994e-7, 8.461904122680
114e-7, 9.647298903318074e-7, 1.082336054532774e-6, 1.198895128502577e-6  …
  1.198895128502574e-6, 1.082336054532771e-6, 9.647298903317973e-7, 8.46190
4122680173e-7, 7.268322996306977e-7, 6.067710236852082e-7, 4.86122735962001
8e-7, 3.650041558874788e-7, 2.4353245786555383e-7, 1.2182515791894544e-7]
 [1.3840375654998628e-8, 2.7667361640417535e-8, 4.146758124031757e-8, 5.522
768363350887e-8, 6.893435680960814e-8, 8.257434044754899e-8, 9.613443874408
605e-8, 1.0960153317988251e-7, 1.2296259521083026e-7, 1.3620469887232483e-7
  …  1.3620469887232541e-7, 1.2296259521083076e-7, 1.0960153317988296e-7, 9
.613443874408645e-8, 8.257434044754931e-8, 6.89343568096084e-8, 5.522768363
350908e-8, 4.1467581240317726e-8, 2.766736164041764e-8, 1.384037565499868e-
8]
 [8.101464381585516e-10, 1.6195091119608046e-9, 2.42730501529131e-9, 3.2327
526577846295e-9, 4.035072820865971e-9, 4.833489311593355e-9, 5.627229713578
58e-9, 6.415526134244419e-9, 7.1976159477151304e-9, 7.972742532607715e-9  …
  7.972742532607265e-9, 7.197615947715326e-9, 6.4155261342441545e-9, 5.6272
29713578173e-9, 4.833489311593434e-9, 4.035072820865746e-9, 3.2327526577846
526e-9, 2.427305015291344e-9, 1.619509111960698e-9, 8.101464381585768e-10]
 [1.5091245482415332e-10, 3.016789115947904e-10, 4.521535135020832e-10, 6.0
2190686086934e-10, 7.516452780748675e-10, 9.003727018005037e-10, 1.04822907
30867924e-9, 1.1950713504436737e-9, 1.340757473451457e-9, 1.485146500195128
7e-9  …  1.4851465001951064e-9, 1.340757473451437e-9, 1.1950713504436539e-9
, 1.0482290730867758e-9, 9.003727018004875e-10, 7.516452780748567e-10, 6.02
1906860869282e-10, 4.521535135020774e-10, 3.016789115947865e-10, 1.50912454
82415115e-10]
 [6.641002153830798e-11, 1.3275579566987847e-10, 1.9897313714290608e-10, 2.
649979849568602e-10, 3.307664643338677e-10, 3.9621494851886006e-10, 4.61280
12033193517e-10, 5.258990334178828e-10, 5.900091731557386e-10, 6.5354851712
15601e-10  …  6.535485171220915e-10, 5.90009173155402e-10, 5.25899033418360
9e-10, 4.612801203315189e-10, 3.962149485196002e-10, 3.307664643338187e-10,
 2.649979849560876e-10, 1.9897313714319766e-10, 1.3275579566986694e-10, 6.6
41002153836793e-11]
 [1.2715036794059667e-11, 2.5417772611208764e-11, 3.809591837493723e-11, 5.
073720879802396e-11, 6.332941424840892e-11, 7.586035258057406e-11, 8.831790
092098079e-11, 1.006900073961674e-10, 1.1296470279215916e-10, 1.25130112133
9091e-10  …  1.251301121339372e-10, 1.1296470279218449e-10, 1.0069000739619
025e-10, 8.831790092100059e-11, 7.586035258059133e-11, 6.332941424842334e-1
1, 5.0737208798035176e-11, 3.809591837494568e-11, 2.5417772611214386e-11, 1
.2715036794062485e-11]
 [2.5767633173290218e-12, 5.151033782531127e-12, 7.72032095529055e-12, 1.02
82139216008035e-11, 1.2834010171267459e-11, 1.5373465050458493e-11, 1.78980
47095047836e-11, 2.040531393508339e-11, 2.2892839951724236e-11, 2.535821862
4275065e-11  …  2.5358218624241513e-11, 2.2892839951773557e-11, 2.040531393
4995165e-11, 1.7898047095136473e-11, 1.53734650504096e-11, 1.28340101712056
53e-11, 1.028213921608327e-11, 7.720320955280572e-12, 5.1510337825353014e-1
2, 2.576763317326595e-12]

Simple Error Analysis

Now let's check the error at an arbitrary timepoint in there. Our analytical solution only exists in the Stratanovich sense, so we are limited in the methods we can calculate errors for.

function simple_error(alg;kwargs...)
    sol = solve(generate_stiff_stoch_heat(1.0,1.0,t_end=0.25),alg;kwargs...);
    sum(abs2,sol[end] - exp(A*sol.t[end]+sol.W[end]*I)*prob.u0)
end
simple_error (generic function with 1 method)
mean(simple_error(EulerHeun(),dt=0.00005) for i in 1:400)
3.2934621489326047e-9
mean(simple_error(ImplicitRKMil(interpretation=:Stratanovich),dt=0.1) for i in 1:400)
0.0023543296132834783
mean(simple_error(ImplicitRKMil(interpretation=:Stratanovich),dt=0.01) for i in 1:400)
0.0031166493582675085
mean(simple_error(ImplicitRKMil(interpretation=:Stratanovich),dt=0.001) for i in 1:400)
0.002256911970940838
mean(simple_error(ImplicitEulerHeun(),dt=0.001) for i in 1:400)
0.0010113378418320853
mean(simple_error(ImplicitEulerHeun(),dt=0.01) for i in 1:400)
0.0010518812009812702
mean(simple_error(ImplicitEulerHeun(),dt=0.1) for i in 1:400)
0.0009781521779762892

Interesting Property

Note that RSwM1 and RSwM2 are not stable on this problem.

sol = solve(generate_stiff_stoch_heat(1.0,1.0,adaptivealg=:RSwM1),SRIW1());
retcode: Success
Interpolation: 1st order linear
t: 91702-element Array{Float64,1}:
 0.0
 1.535027855349674e-5
 1.842033426419609e-5
 2.1874146938732856e-5
 2.575968619758672e-5
 3.0130917863797315e-5
 3.504855348828424e-5
 4.0580893565832026e-5
 4.6804776153073284e-5
 5.38066440637197e-5
 ⋮
 2.9997041717703294
 2.9997304146785626
 2.999759937950325
 2.9997931516310574
 2.9998305170218815
 2.9998725530865586
 2.9999198436593204
 2.9999730455536775
 3.0
u: 91702-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 
1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.8766582974344886, 0.9960098821968079, 1.0077914347803048, 1.00779143478
03048, 1.0077914347803048, 1.0077914347803048, 1.0077914347803048, 1.007791
4347803048, 1.0077914347803048, 1.0077914347803048  …  1.0077914347803048, 
1.0077914347803048, 1.0077914347803048, 1.0077914347803048, 1.0077914347803
048, 1.0077914347803048, 1.0077914347803048, 1.0077914347803048, 0.99600988
21968079, 0.8766582974344886]
 [0.8538978818224252, 0.9922442819476901, 1.0071850678959147, 1.00758073220
38226, 1.0075862844030736, 1.0075862844030736, 1.0075862844030736, 1.007586
2844030736, 1.0075862844030736, 1.0075862844030736  …  1.0075862844030736, 
1.0075862844030736, 1.0075862844030736, 1.0075862844030736, 1.0075862844030
736, 1.0075862844030736, 1.0075807322038226, 1.0071850678959147, 0.99224428
19476901, 0.8538978818224252]
 [0.8325525905391782, 0.9907780360380443, 1.0097687897200955, 1.01070437419
08839, 1.0107312543255376, 1.0107316694302968, 1.0107316727418567, 1.010731
6727418567, 1.0107316727418567, 1.0107316727418567  …  1.0107316727418567, 
1.0107316727418567, 1.0107316727418567, 1.0107316727418567, 1.0107316694302
968, 1.0107312543255376, 1.0107043741908839, 1.0097687897200955, 0.99077803
60380443, 0.8325525905391782]
 [0.8066734018942644, 0.9843335515874002, 1.0082493494190645, 1.00991624864
01345, 1.0099889948815979, 1.0099910482031222, 1.0099910864250057, 1.009991
086856935, 1.0099910868594348, 1.0099910868594348  …  1.0099910868594348, 1
.0099910868594348, 1.009991086856935, 1.0099910864250057, 1.009991048203122
2, 1.0099889948815979, 1.0099162486401345, 1.0082493494190647, 0.9843335515
874002, 0.8066734018942644]
 [0.7807542791960883, 0.9779389258573868, 1.0078450956861396, 1.01050420035
04885, 1.0106605217346853, 1.0106669066599947, 1.0106670932693558, 1.010667
0971538936, 1.01066709720995, 1.0106670972104623  …  1.0106670972104623, 1.
01066709720995, 1.0106670971538936, 1.0106670932693558, 1.0106669066599947,
 1.0106605217346853, 1.0105042003504885, 1.0078450956861398, 0.977938925857
3868, 0.7807542791960883]
 [0.7548853260875761, 0.9714673892681138, 1.0085336677865837, 1.01253027750
18998, 1.0128272926088417, 1.012843279646831, 1.0128439225902897, 1.0128439
421453252, 1.0128439425946971, 1.0128439426023523  …  1.0128439426023523, 1
.0128439425946971, 1.0128439421453252, 1.0128439225902897, 1.01284327964683
1, 1.0128272926088417, 1.0125302775018998, 1.008533667786584, 0.97146738926
81138, 0.7548853260875761]
 [0.7276904712226625, 0.9628195866594308, 1.0082091076734767, 1.01397942436
5642, 1.0145012689784005, 1.0145364468030744, 1.014538269039561, 1.01453834
28209147, 1.0145383451736156, 1.0145383452325876  …  1.0145383452325876, 1.
0145383451736156, 1.0145383428209147, 1.014538269039561, 1.0145364468030744
, 1.0145012689784005, 1.013979424365642, 1.008209107673477, 0.9628195866594
308, 0.7276904712226625]
 [0.698193052576497, 0.9501942611690055, 1.0049687563331122, 1.013040787854
9545, 1.0139075578528731, 1.013978442730141, 1.0139829888933316, 1.01398322
1809238, 1.0139832314458395, 1.0139832317692155  …  1.0139832317692155, 1.0
139832314458395, 1.013983221809238, 1.0139829888933316, 1.013978442730141, 
1.0139075578528731, 1.0130407878549545, 1.0049687563331124, 0.9501942611690
055, 0.698193052576497]
 [0.6677783545946766, 0.9349122797635463, 1.0001096126794524, 1.01112400389
86324, 1.0125050079405262, 1.012639101293439, 1.0126494730130695, 1.0126501
241382138, 1.0126501577229765, 1.0126501591563044  …  1.0126501591563044, 1
.0126501577229765, 1.0126501241382138, 1.0126494730130695, 1.01263910129343
9, 1.0125050079405262, 1.0111240038986324, 1.0001096126794526, 0.9349122797
635463, 0.6677783545946766]
 ⋮
 [-1.613986698050512e-6, 3.226412342311566e-6, -4.835715893161178e-6, 6.440
34232206141e-6, -8.038736642900345e-6, 9.629355868523882e-6, -1.12106570822
3122e-5, 1.2781115307804405e-5, -1.4339205667184427e-5, 1.588342709491269e-
5  …  -1.5883425254684067e-5, 1.4339207331265e-5, -1.278111382703132e-5, 1.
1210658383253846e-5, -9.629354752919822e-6, 8.03873757581847e-6, -6.4403415
75929054e-6, 4.835716454361311e-6, -3.2264119685251934e-6, 1.61398688535904
61e-6]
 [-8.090206333912141e-7, 1.6172589652046892e-6, -2.4239319565084113e-6, 3.2
282610696570873e-6, -4.02946555777335e-6, 4.826773659183178e-6, -5.61940993
9586873e-6, 6.406612399108411e-6, -7.1876139160177755e-6, 7.961665200255427
e-6  …  -7.96166336157097e-6, 7.18761557732499e-6, -6.406610919566914e-6, 5
.619411238432311e-6, -4.826772544500356e-6, 4.0294664891260605e-6, -3.22826
0324137526e-6, 2.423932516764967e-6, -1.6172585917244056e-6, 8.090208203845
116e-7]
 [-4.178352307460396e-7, 8.352666060593037e-7, -1.2518891689009535e-6, 1.66
73017278879234e-6, -2.081099791058997e-6, 2.492886383238858e-6, -2.90225904
24129265e-6, 3.308826545460563e-6, -3.7121900173499245e-6, 4.11196550057248
4e-6  …  -4.111963663822967e-6, 3.7121916762399904e-6, -3.308825067470685e-
6, 2.902260339364231e-6, -2.492885269721739e-6, 2.081100721051587e-6, -1.66
73009831462656e-6, 1.2518897283384384e-6, -8.352662329682912e-7, 4.17835417
46581e-7]
 [-2.3187542091043513e-7, 4.635268891160409e-7, -6.947291817723802e-7, 9.25
2604827491259e-7, -1.1548951703897504e-6, 1.3834144233267104e-6, -1.6105930
939491353e-6, 1.836216202253389e-6, -2.0600599457165965e-6, 2.2819140174079
5e-6  …  -2.2819121872690757e-6, 2.0600615983201376e-6, -1.8362147295803008
e-6, 1.6105943859834808e-6, -1.3834133138136368e-6, 1.1548960968558107e-6, 
-9.252597406845173e-7, 6.947297390881067e-7, -4.6352651736593156e-7, 2.3187
560692196726e-7]
 [-1.445579324914143e-7, 2.889763851802698e-7, -4.331145309150567e-7, 5.768
347768676418e-7, -7.199954933553407e-7, 8.624615068671764e-7, -1.0040909331
014806e-6, 1.1447515408452807e-6, -1.284301740091757e-6, 1.422612753497398e
-6  …  -1.4226109288340547e-6, 1.2843033876026038e-6, -1.1447500725768593e-
6, 1.0040922211533815e-6, -8.624604006717352e-7, 7.199964169655059e-7, -5.7
68340370213887e-7, 4.33115086512699e-7, -2.8897601454136955e-7, 1.445581179
294802e-7]
 [-1.052452608163912e-7, 2.103890762598442e-7, -3.15328608979558e-7, 4.1996
419805786346e-7, -5.241920120096568e-7, 6.279145599621297e-7, -7.3102741797
69395e-7, 8.334356424176697e-7, -9.350346208805973e-7, 1.0357323242799419e-
6  …  -1.0357304899392001e-6, 9.350362770607268e-7, -8.334341663612441e-7, 
7.310287128063304e-7, -6.279134478986982e-7, 5.241929404796918e-7, -4.19963
4542866238e-7, 3.15329167500661e-7, -2.1038870365457645e-7, 1.0524544723019
04e-7]
 [-9.434797220245416e-8, 1.8860504153201954e-7, -2.826789026304864e-7, 3.76
4804068004306e-7, -4.6991620329688135e-7, 5.62899242515557e-7, -6.553354894
60601e-7, 7.471403304147993e-7, -8.382194099273508e-7, 9.284908775188141e-7
  …  -9.284890427912276e-7, 8.382210664380913e-7, -7.471388540466228e-7, 6.
553367845485543e-7, -5.628981302169596e-7, 4.699171319524845e-7, -3.7647966
287167296e-7, 2.8267946126330336e-7, -1.8860466884778368e-7, 9.434815865354
571e-8]
 [-1.0796004009776513e-7, 2.1581600548745907e-7, -3.23462443870237e-7, 4.30
7970607777275e-7, -5.377134342833095e-7, 6.441114459320609e-7, -7.498841158
780861e-7, 8.549338890324049e-7, -9.591536415078176e-7, 1.0624487590003023e
-6  …  -1.0624469392480143e-6, 9.591552845223746e-7, -8.549324247143803e-7,
 7.498854004149139e-7, -6.441103427119498e-7, 5.377143553734327e-7, -4.3079
632292072166e-7, 3.2346299795218963e-7, -2.1581563584493686e-7, 1.079602250
2996661e-7]
 [-5.429191461851022e-8, 1.0853167553916458e-7, -1.6266569898401866e-7, 2.1
664346311171126e-7, -2.704101613586216e-7, 3.2391709921439417e-7, -3.771084
5934956773e-7, 4.299375629870953e-7, -4.823477979722086e-7, 5.3429468184983
65e-7  …  -5.342928592675592e-7, 4.823494434502864e-7, -4.2993609639091896e
-7, 3.7710974581198383e-7, -3.2391599427736457e-7, 2.70411083829307e-7, -2.
1664272410609863e-7, 1.6266625389638003e-7, -1.085313053211375e-7, 5.429209
982785935e-8]

Conclusion

In this problem, the implicit methods do not have a stepsize limit. This is because the stiffness almost entirely deteriministic due to diffusion. In that case, if we do not care about the error too much, the implicit methods dominate. Of course, as the tolerance gets lower there is a tradeoff point where the higher order methods will become more efficient. The explicit methods are clearly stability-bound and thus unless we want an error of like 10^-10 we are better off using an implicit method here.

using DiffEqBenchmarks
DiffEqBenchmarks.bench_footer(WEAVE_ARGS[:folder],WEAVE_ARGS[:file])

Appendix

These benchmarks are a part of the DiffEqBenchmarks.jl repository, found at: https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl

To locally run this tutorial, do the following commands:

using DiffEqBenchmarks
DiffEqBenchmarks.weave_file("StiffSDE","StochasticHeat.jmd")

Computer Information:

Julia Version 1.4.2
Commit 44fa15b150* (2020-05-23 18:35 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
Environment:
  JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqBenchmarks.jl/.julia
  JULIA_CUDA_MEMORY_LIMIT = 2147483648
  JULIA_PROJECT = @.
  JULIA_NUM_THREADS = 8

Package Information:

Status: `/builds/JuliaGPU/DiffEqBenchmarks.jl/benchmarks/StiffSDE/Project.toml`
[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.22.0
[77a26b50-5914-5dd7-bc55-306e6241c503] DiffEqNoiseProcess 5.0.2
[a077e3f3-b75c-5d7f-a0c6-6bc4c8ec64a9] DiffEqProblemLibrary 4.8.0
[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.3
[789caeaf-c7a9-5a7d-9973-96adeb23e2a0] StochasticDiffEq 6.24.0
[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra 
[9a3f8284-a2c9-5f02-9a11-845980a1fd5c] Random 
[10745b16-79ce-11e8-11f9-7d13ad32a3b2] Statistics