Solves the Kuramoto-Sivashinsky equation
\begin{equation}
u_t +uu_x= -d u_{xx}-\nu u_{xxxx}
\end{equation}
in Fourier space using ApproxFun.jl and DifferentialEquations.jl. The negative diffusivity and hyperviscosity can be compared to find the diffusive cutoff scale. Typically, larger the $d$ or smaller the $\nu$, more number of unstable modes manifest.

Currently numerically unstable for $\nu$=0.2 and $d$<0.02. It should be stable for All values of $d$ and $\nu$ for sufficiently small values of $dt$. One check is that the ApproxFun derivatives are correct, especially for high wavenumbers.


```julia
using DifferentialEquations
using ApproxFun, Sundials
using PyPlot
```


```julia
#using Pkg
#Pkg.add("DiffEqOperators")
using DiffEqOperators, LinearAlgebra
```


```julia
S = Fourier()
#S = Chebyshev()
n = 2^8
x = points(S, n)
T = ApproxFun.plan_transform(S, n)
Ti = ApproxFun.plan_itransform(S, n)

d = 0.2
nu = 2e-2
dt = 1e-3
Nmax =  (3000+1)
t0 = Nmax * dt

u₀ = T*(cos.(x).*(sin.(x).+1));
```


```julia

# D1 = Derivative(S,1)
# D2 = Derivative(S,2)
# D4 = Derivative(S,4)
# L1 = D1[1:n,1:n];
# L2 = D2[1:n,1:n];
# L4 = D4[1:n,1:n];

```


```julia
function d_dx(S, n, N)
    D = Derivative(S,n)
    return D[1:N,1:N]    
end    
```




    d_dx (generic function with 1 method)




```julia
Dx(S,m) = d_dx(S,m,n)
L1 = Dx(S,1)
L2 = Dx(S,2)
L4 = Dx(S,4)
```




    256×256 BandedMatrices.BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
     0.0   ⋅    ⋅     ⋅     ⋅     ⋅     ⋅   …   ⋅          ⋅          ⋅ 
      ⋅   1.0   ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅   1.0    ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅   16.0    ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅   16.0    ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅   81.0    ⋅   …   ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅   81.0      ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅   …   ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
     ⋮                           ⋮          ⋱                        ⋮
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅   …   ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅   …   ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅          ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅      2.60145e8   ⋅          ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅       ⋅         2.60145e8   ⋅ 
      ⋅    ⋅    ⋅     ⋅     ⋅     ⋅     ⋅   …   ⋅          ⋅         2.68435e8




```julia
A = DiffEqArrayOperator(-d*Diagonal(L2)-nu*Diagonal(L4))
```




    DiffEqArrayOperator{Float64,Diagonal{Float64,Array{Float64,1}},typeof(DiffEqBase.DEFAULT_UPDATE_FUNC)}([-0.0 0.0 … 0.0 0.0; 0.0 0.18000000000000002 … 0.0 0.0; … ; 0.0 0.0 … -5.1996670200000005e6 0.0; 0.0 0.0 … 0.0 -5.36543232e6], DiffEqBase.DEFAULT_UPDATE_FUNC)




```julia
function f(dû,û,tmp,t)
  # Transform u back to point-space
  mul!(tmp,Ti,û)
  tmp= 0.5.*tmp.*tmp
  mul!(tmp, T, tmp)
  mul!(dû,-L1,tmp)    
  #mul!(dû,T,tmp) # Transform back to Fourier space
end
```




    f (generic function with 1 method)




```julia
prob = SplitODEProblem(A, f, u₀, (0.0,t0), similar(u₀));
```


```julia
sol = solve(prob, ETDRK4(), dt=dt)
```




    retcode: Success
    Interpolation: 3rd order Hermite
    t: 3004-element Array{Float64,1}:
     0.0
     0.001
     0.002
     0.003
     0.004
     0.005
     0.006
     0.007
     0.008
     0.009000000000000001
     0.010000000000000002
     0.011000000000000003
     0.012000000000000004
     ⋮
     2.9919999999997815
     2.9929999999997814
     2.9939999999997813
     2.994999999999781
     2.995999999999781
     2.996999999999781
     2.997999999999781
     2.9989999999997807
     2.9999999999997806
     3.0009999999997805
     3.0019999999997804
     3.001
    u: 3004-element Array{Array{Float64,1},1}:
     [-5.2512003524312095e-17, 9.412373427195134e-17, 1.0, 0.5, -1.0285899315265006e-16, 5.872110630286023e-17, -1.931231632089504e-17, 6.099563844139619e-17, 3.49507587983103e-18, 3.0041826889091146e-17  …  -3.2246697248208723e-17, -6.197698520320395e-18, -1.4326908838472506e-17, -5.003201408188741e-18, -5.235528008911165e-18, -0.0, -8.316563711914074e-18, 2.1534186447614525e-18, 0.0, 7.571823474787936e-18]
     [-5.2590274950011845e-17, 9.414067806901627e-17, 1.000680088840932, 0.4999898837913804, -1.0312136357202911e-16, 5.865030686229162e-17, -0.00025026703177348557, -0.0007489657447018696, 3.544423735217371e-18, 2.979811297280835e-17  …  1.8804149443628929e-22, 1.210926565912341e-22, -2.667250985753466e-23, -3.8037668574224798e-22, -5.957337461161826e-31, -3.4454825974466027e-22, -6.463952756579517e-23, 4.807397683757782e-23, 0.0, 8.593557279146788e-31]
     [-5.266548456843873e-17, 9.415762491623919e-17, 1.0013603559259554, 0.4999795349309591, -1.0336238430621092e-16, 5.858187354674103e-17, -0.0005010662726383619, -0.0014958659554359436, 3.572327876964712e-18, 2.953460336887781e-17  …  3.7583642255593657e-22, 2.0846054662299754e-22, 4.746693894812992e-22, 1.1679246957166403e-22, -6.787491182499563e-44, 1.3665034426231597e-25, 2.262606427213538e-22, -6.739347426280095e-23, 0.0, -6.566083506531838e-22]
     [-5.27421629965902e-17, 9.417457481416916e-17, 1.002040802095086, 0.49996895306670597, -1.0362085626177933e-16, 5.852308628310149e-17, -0.0007523949820173098, -0.0022407050922477255, 3.607314507901586e-18, 2.9289525405986584e-17  …  1.8808261170014646e-22, -1.9165622814373872e-22, 2.131768373327274e-22, -6.597013380134723e-23, -7.163921344431885e-22, -6.88345078275293e-22, 4.4292866328970035e-23, -5.764495672323714e-23, 0.0, -6.119560250402797e-25]
     [-5.282850605975103e-17, 9.419152776335535e-17, 1.002721428184002, 0.49995813784634624, -1.0387108133803322e-16, 5.846927381877495e-17, -0.0010042504734009594, -0.002983487608670593, 3.6509067023077235e-18, 2.906194037564697e-17  …  -1.6438244782106967e-25, 2.8172017742467005e-23, -5.82746977877344e-24, 2.1323394884481013e-22, -8.152716454484188e-35, -3.4159872490536635e-25, 2.042116825286874e-23, 2.056231172925457e-22, 0.0, -2.4480066700185824e-25]
     [-5.2914884199119876e-17, 9.420848376434706e-17, 1.0034022350241596, 0.4999470889173666, -1.0414431482449414e-16, 5.84040086291147e-17, -0.0012566301134442914, -0.0037242179517453272, 3.683261602622675e-18, 2.8842777571695694e-17  …  3.7616522340020725e-22, -2.2021301700757697e-22, -1.5195912376552988e-23, -3.5695749460913856e-23, -2.9362208709026728e-25, -6.886866770001204e-22, 2.5909673782143266e-22, 1.5070183468333224e-22, 0.0, -9.128507753042536e-29]
     [-5.2992712908932524e-17, 9.422544281769365e-17, 1.0040832234429051, 0.49993580592702114, -1.0442714650730606e-16, 5.832806046592029e-17, -0.0015095313210832897, -0.004462900562038755, 3.694267555967532e-18, 2.859850032506395e-17  …  -2.8765130937799638e-25, -2.1996226122328675e-22, 1.4918730882845223e-22, -3.001374969203127e-22, 7.16245263202657e-22, -3.4407006494746153e-22, 2.0493408253477608e-23, -1.1819253130376407e-22, 0.0, -1.0363584023153932e-41]
     [-5.3074258552181076e-17, 9.42424049239446e-17, 1.0047643942635824, 0.49992428852233733, -1.0470061614994359e-16, 5.8263336768881e-17, -0.0017629515666711965, -0.00519953987366149, 3.720326750644124e-18, 2.837547247806459e-17  …  -3.594860711715409e-29, 3.954124902735485e-23, -3.5531084843585534e-22, -5.105212235199744e-22, 5.872441742619519e-25, -3.440017126393458e-22, -1.6267086396184128e-23, -1.6007447751402034e-22, 0.0, -1.1754097471692263e-54]
     [-5.31624449337361e-17, 9.425937008364948e-17, 1.0054457483056378, 0.49991253635012245, -1.0495957498988596e-16, 5.820591525429658e-17, -0.0020168883711337454, -0.005934140314284811, 3.75525276013132e-18, 2.815992260939632e-17  …  1.8783603802833986e-22, -1.8603192058676038e-22, -7.171431428646145e-23, -4.2000648733057975e-22, -1.4330776501855222e-21, -2.0494838056501831e-25, -2.879924630272045e-22, -4.46357139841468e-22, 0.0, -1.3356376419092575e-67]
     [-5.324305040316962e-17, 9.427633829735797e-17, 1.0061272863847213, 0.4999005490569699, -1.0522283373475682e-16, 5.813957890500789e-17, -0.0022713393051427733, -0.006666706305156714, 3.779687461045498e-18, 2.7911357177487176e-17  …  -4.9311139489902685e-25, 4.397060832045062e-22, -8.499471629484644e-24, -1.3067699209543713e-22, -7.163921344433516e-22, -3.4427506759976148e-22, -7.376230124278615e-23, -1.498163868097151e-22, 0.0, -1.3130943466154056e-21]
     [-5.332475353995231e-17, 9.429330956561981e-17, 1.0068090093127844, 0.49988832628926616, -1.0547902418413103e-16, 5.807372593515286e-17, -0.0025263019883076587, -0.007397242261117177, 3.815907411341094e-18, 2.7711632466806366e-17  …  3.7628849226178553e-22, -1.241092406749261e-22, -2.9242352644912965e-22, 6.596167276301269e-23, -2.936220871717899e-25, -6.829803626369174e-26, -4.644094198102284e-22, 1.3065256698729292e-23, 0.0, -4.89601334152689e-25]
     [-5.3405212146666403e-17, 9.431028388898489e-17, 1.007490917898174, 0.4998758676931971, -1.0574191621406768e-16, 5.801730339529358e-17, -0.0027817740883840474, -0.008125752590612666, 3.857209732275829e-18, 2.7491422995233512e-17  …  -7.517961858705798e-22, -6.752879321445586e-23, -1.3604316657412998e-22, 4.308869878396189e-22, -3.342788959246766e-38, -1.3665034430135613e-25, 1.4981472312345496e-22, 3.046342380757894e-22, 0.0, 6.562411952952198e-22]
     [-5.3483376390714264e-17, 9.432726126800317e-17, 1.0081730129457243, 0.4998631729147552, -1.0600780968041389e-16, 5.795085230510223e-17, -0.003037753320499366, -0.008852241695709907, 3.895119351495902e-18, 2.7289011378087923e-17  …  -1.8795930689000397e-22, 6.763622112323714e-23, -2.8857668555660514e-22, -5.4820157137698325e-24, -1.4321969043182241e-21, -3.4400171263930683e-22, 2.4503176296518646e-22, 2.135429328507511e-22, 0.0, 7.447018283862937e-35]
     ⋮
     [-6.560457939484194e-15, 1.612845568431533e-16, -0.3647415103398549, 0.24635607535895176, 3.044171002241246e-14, -7.671244327306749e-15, -1.0585724211413072, 0.18757718170484242, -2.5872832767206335e-14, -1.8678427024247974e-15  …  -3.757542313320796e-22, -7.638167529233835e-26, -2.101447026240043e-22, -5.729377095904904e-22, 3.582693824478874e-22, 6.835230807664333e-26, 2.1575889562034337e-22, 4.475198977593116e-23, 0.0, -1.2297637107759614e-22]
     [-6.570029711118277e-15, 1.6131359067635164e-16, -0.36452371986258103, 0.24650224573296428, 3.047352728602491e-14, -7.685506522212395e-15, -1.0581210813637953, 0.1875951236208701, -2.58884761020377e-14, -1.8710066600402025e-15  …  -3.7591861377987915e-22, -1.5276335058465064e-25, 3.7612302425340136e-22, 3.020694131236111e-23, 3.584161934914733e-22, 3.440017669111588e-22, 3.6572657460062107e-22, -4.0140237370509236e-23, 0.0, 4.106096342036543e-23]
     [-6.579611710083211e-15, 1.6134262973611034e-16, -0.3643060077663784, 0.24664841934121046, 3.050535320396346e-14, -7.699802068761142e-15, -1.0576703614337943, 0.1876129274172492, -2.590410753224542e-14, -1.874182180543233e-15  …  -7.518372275597156e-22, -1.5282806227753347e-25, 2.1891070142232236e-22, -6.912197585853458e-22, -8.808060642437212e-25, 6.835230811564699e-26, 2.5626507329298564e-22, 9.019457984473498e-23, 0.0, -1.2300698257160606e-22]
     [-6.589200101190471e-15, 1.6137167402337027e-16, -0.3640883742418364, 0.24679459613455842, 3.053722900891118e-14, -7.71409132343964e-15, -1.0572202607261127, 0.18763059328802273, -2.5919723377474052e-14, -1.877363917956123e-15  …  -7.520838371801294e-22, 3.614652571219553e-22, 7.73874317912191e-23, -2.365003591704265e-22, -7.1661232091001325e-22, 6.882767259672164e-22, 1.643591467904741e-22, 2.6499871603991594e-22, 0.0, 1.2321350746045147e-22]
     [-6.598799787487542e-15, 1.6140072353907246e-16, -0.36387081947835637, 0.24694077606403275, 3.056913426094282e-14, -7.72840601833286e-15, -1.0567707786162919, 0.1876481214272999, -2.5935352000036173e-14, -1.8805424305279135e-15  …  8.215527522844704e-26, 3.6115966570913427e-22, -2.641283916638285e-23, 5.204274305363828e-22, 2.201864667430097e-25, 2.733006886806292e-25, -5.400048669387192e-22, 1.6198910340593646e-22, 0.0, -8.193833204581079e-23]
     [-6.6084084993687495e-15, 1.6142977828415812e-16, -0.36365334366415764, 0.24708695908081407, 3.0601067993264706e-14, -7.742754186884353e-15, -1.0563219144806064, 0.18766551202925513, -2.595103063506318e-14, -1.883662334853968e-15  …  -3.75836386607308e-22, 7.219372289066872e-22, -6.472598898723793e-23, 5.745678333968437e-23, -2.935618901039587e-25, 6.884133763115177e-22, 6.51835552850291e-23, -1.5051357131665847e-22, 0.0, 1.530004167830771e-26]
     [-6.618031021695917e-15, 1.6145883825956858e-16, -0.3634359469862827, 0.24723314513623848, 3.0633037040879555e-14, -7.75709396798161e-15, -1.0558736676960645, 0.18768276528812697, -2.5966702147679024e-14, -1.8868623932188247e-15  …  8.21552752713487e-26, -2.2920973748765473e-25, 1.452556961935484e-22, -3.855842397491083e-22, 2.2018646682452524e-25, 3.4413836298368627e-22, -2.3028549516554434e-22, 1.7908996906754248e-23, 0.0, 4.09921274960785e-23]
     [-6.627664044242447e-15, 1.6148790346624542e-16, -0.363218629630603, 0.24737933418179725, 3.0665037393654965e-14, -7.771470022897515e-15, -1.055426037640408, 0.18769988139821714, -2.5982385379322564e-14, -1.8899989739099774e-15  …  -3.2876489568500755e-25, 4.5848418683252215e-25, 1.8995477275311838e-22, -3.6913137055227007e-22, -3.5841625368841884e-22, 6.829803626366918e-26, 1.5924540816215731e-22, 1.13306687820326e-22, 0.0, -8.218312700750459e-23]
     [-6.637305502107754e-15, 1.6151697390513037e-16, -0.36300139178182406, 0.24752552616913615, 3.069705181155621e-14, -7.78585577406177e-15, -1.054979023692112, 0.18771686055388956, -2.5998069582521562e-14, -1.8931236797535567e-15  …  -2.466096203708845e-25, 5.217579895899584e-38, 1.6450198776007582e-22, 4.515230071943567e-22, 2.2018646678377103e-25, 7.752617390544438e-39, -4.032794184184933e-22, 7.561422331481542e-23, 0.0, 2.0515949808510068e-22]
     [-6.646969258131776e-15, 1.6154604957716528e-16, -0.36278423362349155, 0.24767172105005508, 3.072909854494172e-14, -7.800253707139348e-15, -1.0545326252303857, 0.18773370294956931, -2.6013779133013582e-14, -1.8963058070771215e-15  …  -3.764939163986781e-22, 1.5282806227751607e-25, 2.7797597081822365e-22, -5.371014659332686e-22, -7.166856963332926e-22, 1.366503443013484e-25, 1.6507529725439735e-22, -1.2883944554277081e-22, 0.0, -8.203014941187923e-23]
     [-6.656639717563624e-15, 1.615751304832922e-16, -0.36256715533799616, 0.24781791877650763, 3.076118340269337e-14, -7.814665097096839e-15, -1.0540868416351723, 0.1877504087797416, -2.6029500866058312e-14, -1.89949783661429e-15  …  -3.758364585045652e-22, 1.5282806227753345e-25, 1.4122725538244355e-22, 1.098107397016619e-22, -3.581225714044239e-22, -1.366503443013329e-25, 2.113593320888721e-24, 7.624740972989727e-23, 0.0, -8.214489116659288e-23]
     [-6.646969258133894e-15, 1.6154604957717164e-16, -0.3627842336234439, 0.24767172105008717, 3.072909854494876e-14, -7.80025370714251e-15, -1.054532625230288, 0.187733702949573, -2.601377913301702e-14, -1.8963058070778267e-15  …  -3.7649391606880204e-22, 1.528279038485104e-25, 2.77976085216198e-22, -5.371005134435813e-22, -7.166841619002893e-22, 1.3665019327718588e-25, 1.65075296176036e-22, -1.288392944456508e-22, 0.0, -8.203000452514313e-23]




```julia
plot(Ti*sol(0.001))
plot(Ti*sol(3))
#plot(Ti*sol(3))

```


```julia
#
u = zeros(eltype(sol),(n, Nmax))
for it in 1:Nmax
   print(dt*it)
   u[:,it] = Ti*sol(dt*it)
   #println(u[:,it]) 
end
size(u)
```

    0.0010.0020.0030.0040.0050.0060.0070.0080.0090000000000000010.010.0110.0120.0130000000000000010.0140.0150.0160.0170.0180000000000000020.0190.020.0210.0220.0230.0240.0250.0260000000000000020.0270.0280.0290.030.0310.0320.0330.0340.0350.0360000000000000040.0370.0380.0390.040.0410.0420.0430000000000000030.0440.0450.0460.0470.0480.0490.050.0510000000000000040.0520000000000000050.0530.0540.0550.0560.0570.0580.0590000000000000040.060.0610.0620.0630.0640.0650.0660.0670.0680.0690.070.071000000000000010.072000000000000010.0730.0740.0750.0760.0770.0780.0790.080.0810.0820.0830.0840.0850.086000000000000010.087000000000000010.0880.0890.090.0910.0920.0930.0940.0950.0960.0970.0980.0990.10.1010.102000000000000010.103000000000000010.104000000000000010.1050.1060.1070.1080.1090.110.1110.1120.1130.1140.1150.1160.1170.118000000000000010.119000000000000010.120.1210.1220.1230.1240.1250.1260.1270.1280.1290.130.1310.1320.1330.1340.1350.1360.1370.1380.1390.140.141000000000000010.142000000000000020.143000000000000020.144000000000000020.1450.1460.1470.1480.1490.150.1510.1520.1530.1540.1550.1560.1570.1580.1590.160.1610.1620.1630.1640.1650.1660.1670.1680.1690.170.1710.172000000000000010.173000000000000010.174000000000000020.175000000000000020.1760.1770.1780.1790.180.1810.1820.1830.1840.1850.1860.1870.1880.1890.190.1910.1920.1930.1940.1950.1960.1970.1980.1990.20.2010.2020.2030.204000000000000010.205000000000000020.206000000000000020.207000000000000020.208000000000000020.2090.210.2110.2120.2130.2140.2150.2160.2170.2180.2190.220.2210.2220.2230.2240.2250.2260.2270.2280.2290.230.2310.2320.2330.2340.235000000000000010.236000000000000020.237000000000000020.238000000000000020.239000000000000020.240.2410.2420.2430.2440.2450.2460.2470.2480.2490.250.2510.2520.2530.2540.2550.2560.2570.2580.2590.260.2610.2620.2630.2640.2650.2660.2670.2680.2690.270.2710.2720.2730.2740.2750.2760.2770.2780.2790.280.2810.282000000000000030.283000000000000030.284000000000000030.285000000000000030.286000000000000030.287000000000000030.288000000000000030.2890.290.2910.2920.2930.2940.2950.2960.2970.2980.2990.30.3010.3020.3030.3040.3050.3060.3070.3080.3090.310.3110.3120.3130.3140.3150.3160.3170.3180.3190.320.3210.3220.3230.3240.3250.3260.3270.3280.3290.330.3310.3320.3330.3340.3350.3360.3370.3380.3390.340.3410.3420.3430.344000000000000030.345000000000000030.346000000000000030.347000000000000030.348000000000000030.349000000000000030.350000000000000030.351000000000000030.3520.3530.3540.3550.3560.3570.3580.3590.360.3610.3620.3630.3640.3650.3660.3670.3680.3690.370.3710.3720.3730.3740.3750.3760.3770.3780.3790.380.3810.3820.3830.3840.3850.3860.3870.3880.3890.390.3910.3920.3930.3940.3950.3960.3970.3980.3990.40.4010.4020.4030.4040.4050.4060.407000000000000030.408000000000000030.409000000000000030.410000000000000030.411000000000000030.412000000000000030.413000000000000030.414000000000000030.415000000000000040.416000000000000040.4170.4180.4190.420.4210.4220.4230.4240.4250.4260.4270.4280.4290.430.4310.4320.4330.4340.4350.4360.4370.4380.4390.440.4410.4420.4430.4440.4450.4460.4470.4480.4490.450.4510.4520.4530.4540.4550.4560.4570.4580.4590.460.4610.4620.4630.4640.4650.4660.4670.4680.469000000000000030.470000000000000030.471000000000000030.472000000000000030.473000000000000030.474000000000000030.475000000000000030.476000000000000030.477000000000000040.478000000000000040.479000000000000040.480.4810.4820.4830.4840.4850.4860.4870.4880.4890.490.4910.4920.4930.4940.4950.4960.4970.4980.4990.50.5010.5020.5030.5040.5050.5060.5070.5080.5090.510.5110.5120.5130.5140.5150.5160.5170.5180.5190.520.5210.5220.5230.5240.5250.5260.5270.5280.5290.530.5310.5320.5330.5340.5350.5360.5370.5380.5390.540.5410.5420.5430.5440.5450.5460.5470.5480.5490.550.5510.5520.5530.5540.5550.5560.5570.5580.5590.560.5610.5620.56300000000000010.56400000000000010.56500000000000010.56600000000000010.56700000000000010.56800000000000010.56900000000000010.57000000000000010.57100000000000010.57200000000000010.57300000000000010.57400000000000010.57500000000000010.57600000000000010.5770.5780.5790.580.5810.5820.5830.5840.5850.5860.5870.5880.5890.590.5910.5920.5930.5940.5950.5960.5970.5980.5990.60.6010.6020.6030.6040.6050.6060.6070.6080.6090.610.6110.6120.6130.6140.6150.6160.6170.6180.6190.620.6210.6220.6230.6240.6250.6260.6270.6280.6290.630.6310.6320.6330.6340.6350.6360.6370.6380.6390.640.6410.6420.6430.6440.6450.6460.6470.6480.6490.650.6510.6520.6530.6540.6550.6560.6570.6580.6590.660.6610.6620.6630.6640.6650.6660.6670.6680.6690.670.6710.6720.6730.6740.6750.6760.6770.6780.6790.680.6810.6820.6830.6840.6850.6860.6870.68800000000000010.68900000000000010.69000000000000010.69100000000000010.69200000000000010.69300000000000010.69400000000000010.69500000000000010.69600000000000010.69700000000000010.69800000000000010.69900000000000010.70000000000000010.70100000000000010.70200000000000010.70300000000000010.7040.7050.7060.7070.7080.7090.710.7110.7120.7130.7140.7150.7160.7170.7180.7190.720.7210.7220.7230.7240.7250.7260.7270.7280.7290.730.7310.7320.7330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    (256, 3001)




```julia

IM = PyPlot.imshow(abs.(u))
cb = PyPlot.colorbar(IM, orientation="horizontal") 
#gcf()
```

8910cc2746c7f3848ddb339ddb8b61b6dc8bacd2
ipynb
Jupyter Notebook
KS_approx.ipynb
aetherks/ksequation
2b43465685c8ba730b69051d8e6ea36349473253
MIT
2020-10-12T00:05:02.000Z
2020-10-12T00:05:02.000Z
KS_approx.ipynb
aetherks/ksequation
2b43465685c8ba730b69051d8e6ea36349473253
MIT
KS_approx.ipynb
aetherks/ksequation
2b43465685c8ba730b69051d8e6ea36349473253
MIT
Qwen/Qwen-72B
1. YES
2. YES