Issue solving 2D wave equation with ModelingToolkit.jl

Specific Domains Modelling & Simulations
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6

I also converted the NeuralODE.jl example into a purely ModelingToolkit.jl script and it looks like a solution to a diffusion problem rather than a wave problem.

This makes me believe its a problem with ModelingToolkit.jl or possibly a boundary condition. I say the latter may be an issue since for a numerical solution to the 2D wave equation I wrote using a spectral method, I saw a similar issue when the initial condition for \frac{du(0,x)}{dx} was improperly defined.

1Dwave

Converted NeuralODE.jl Example

using Plots, DifferentialEquations, ModelingToolkit, DiffEqOperators

@parameters t, x
@variables u(..)
Dxx = Differential(x)^2
Dtt = Differential(t)^2
Dt = Differential(t)
Dx = Differential(x)

#2D PDE
C=1
eq  = Dtt(u(t,x)) ~ C^2*Dxx(u(t,x))

# Initial and boundary conditions
bcs = [u(t,0) ~ 0.,# for all t > 0
       u(t,1) ~ 0.,# for all t > 0
       u(0,x) ~ x*(1. - x), #for all 0 < x < 1
       Dt(u(0,x)) ~ 0.0, #for all  0 < x < 1
]

# Space and time domains
domains = [t ∈ (0.0,1.0),
           x ∈ (0.0,1.0)]
# Method of lines discretization
dx = 0.1
order = 2
discretization = MOLFiniteDifference([x=>dx],t)

# PDE system
pdesys = PDESystem(eq,bcs,domains,[t,x],[u(t,x)])


# Convert the PDE problem into an ODE problem
prob = discretize(pdesys,discretization)

# Solve ODE problem
sol = solve(prob)

# Plot results
anim = @animate for i ∈ 1:length(sol.t)
plot(sol.u[i], label = "wave", ylims =[-0.25, 0.25])
end every 5

gif(anim, "1Dwave.gif", fps = 10)