Study SH model

New to Julia
,
1

Hi everyone,

I’m studying the Swift Hohenberg model with only the linear term of the potential. I see that a tutorial is proposed here (https://rveltz.github.io/BifurcationKit.jl/dev/tutorials2/#Snaking-in-the-2d-Swift-Hohenberg-equation-1) but for the moment I’m not interested to analyze the bifurcation, I wish only to solve the model. My idea was to use DifferentialEquations.jl to solve the equation, but maybe I’ve made some mistakes…
Here my definition of the problem:

function F_sh(u, p)
	@unpack l, L1 = p
	return -L1 * u .+ (l .* u)
end
X = -lx .+ 2lx/(Nx) * collect(0:Nx-1)
Y = -ly .+ 2ly/(Ny) * collect(0:Ny-1)



# define parameters for the PDE
Δ, _ = Laplacian2D(Nx, Ny, lx, ly)
D = 10
#L1 = D*(I + Δ)^2

p = (l = -0.1 , L1 = D*(I + Δ)^2)

u0=rand(100,100)
tspan = (0.0,100.0)
prob = ODEProblem(F_sh,u0,tspan)

sol =  solve(prob, saveat =[0.2,25.0,50.0,99.0])

(I used the Laplacian construction with sparse matrix used in the SH tutorial).
Thank you for the help, I’m very new to Julia, so every comment is welcome!

2

without running it, I’d say you have to pass a flat vector because of the sparse matrix:

prob = ODEProblem(F_sh,vec(u0),tspan)

In passing, you’d better be using a SplitProblem I think

3

Thank you so much,
I tried using the flat vector vec(u0), but it return the same error as before:

MethodError: no method matching F_sh(::Matrix{Float64}, ::SciMLBase.NullParameters, ::Float64)
Closest candidates are:
F_sh(::Any, ::Any) at In[4]:1

I tried also with SplitODEProblem, splicing F_sh into two functions sh1 and sh2, but still the error is returned:

MethodError: no method matching sh1(::Matrix{Float64}, ::SciMLBase.NullParameters, ::Float64)

4

Ah, try:

function F_sh(u, p, t = 0)
	@unpack l, L1 = p
	return -L1 * u .+ (l .* u)
end
5

I’ve already tried this, but the error still remain.
I put here the complete notebook, maybe it could be useful.
Best regards

Francesco

6

At least it works but slowly because of bad solver options


function F_sh(u, p, t)
	@unpack l, L1 = p
	return -L1 * u .+ (l .* u)
end
X = -lx .+ 2lx/(Nx) * collect(0:Nx-1)
Y = -ly .+ 2ly/(Ny) * collect(0:Ny-1)


# define parameters for the PDE
Δ, _ = Laplacian2D(Nx, Ny, lx, ly)
D = 10
#L1 = D*(I + Δ)^2

p = (l = -0.1 , L1 = D*(I + Δ)^2)

u0=rand(Nx,Ny) |> vec
tspan = (0.0,1.0)
prob = ODEProblem(F_sh,u0,tspan, p)

sol =  solve(prob, Tsit5(), saveat =[0.2,25.0,50.0,99.0])
7

I did the same, but still the error is returned… I really don’t understand what I’m missing.

using DifferentialEquations
using DiffEqOperators, Setfield, Parameters
using BifurcationKit, LinearAlgebra, Plots, SparseArrays
const BK = BifurcationKit

# helper function to plot solution
heatmapsol(x) = heatmap(reshape(x,Nx,Ny)',color=:viridis)

Nx = 151
Ny = 100
lx = 4*2pi
ly = 2*2pi/sqrt(3)

# we use DiffEqOperators to compute the Laplacian operator
function Laplacian2D(Nx, Ny, lx, ly)
	hx = 2lx/Nx
	hy = 2ly/Ny
	D2x = CenteredDifference(2, 2, hx, Nx)
	D2y = CenteredDifference(2, 2, hy, Ny)
	Qx = Neumann0BC(hx)
	Qy = Neumann0BC(hy)
	
	A = kron(sparse(I, Ny, Ny), sparse(D2x * Qx)[1]) + kron(sparse(D2y * Qy)[1], sparse(I, Nx, Nx))
	return A, D2x
end

function F_sh(u, p, t)
	@unpack l, L1 = p
	return -L1 .* u .+ (l .* u)
end

X = -lx .+ 2lx/(Nx) * collect(0:Nx-1)
Y = -ly .+ 2ly/(Ny) * collect(0:Ny-1)



# define parameters for the PDE
Δ, _ = Laplacian2D(Nx, Ny, lx, ly)
D = 10
#L1 = D*(I + Δ)^2

p = (l= -0.1, L1 = D*(I + Δ)^2);

u0=rand(Nx,Ny) |> vec
tspan = (0.0,100.0)
prob = ODEProblem(F_sh,u0,tspan,p)
sol =  solve(prob, Tsit5(), saveat =[0.2,50.0,99.0])

MethodError: no method matching Vector{Float64}(::SparseMatrixCSC{Float64, Int64})