Issue solving 2D wave equation with ModelingToolkit.jl

[ChrisRackauckas]

@apattyn

The NeuralODE.jl example looks like the initial waveform just oscillates between ~0.2 and ~-0.2 without ever propagating outwards. Whereas the matlab wave solver shows the initial waveform moving to the boundaries where it is ultimately absorbed my the perfectly-matched-layer (since that toolbox is using a spectral method to solve the wave equation).

The question I suppose would be am I missing something? e.g. I defined the wave equation incorrectly. Or is ModelingToolkit.jl/ NeuralODE.jl giving an incorrect solution for the 2D wave equation?

It looks like it’s the right solution to 2 digits. Plot the slides as an animation if you’re not sure.

@apattyn

MOLFiniteDifference([x=>dx],t)

The documentation on this says it’s not complete.

Note: This uses a currently unreleased interface that is still a work in progress. Use at your own risk!

Specifically, upwinding has not merged yet:

[WIP] Progress on upwindig rules (issue 355)

So yes, it’s as incorrect as the documentation page says it will be right now. We will remove that note when it’s released, but for right now, I mean, the docs said it’s not fully working yet .

(Specifically, as of today, arbitrary even derivatives work fine, it’s the odd ones that need that to merge)

But anyways, @luraess mentioned, if you have a linear PDE like this then the most efficient thing to do would be to semi-discretize it with ParallelStencil.jl and then solve the ODEs using whichever method (likely one of the SSP integrators like https://diffeq.sciml.ai/stable/solvers/ode_solve/#Explicit-Strong-Stability-Preserving-Runge-Kutta-Methods-for-Hyperbolic-PDEs-(Conservation-Laws)). The coming DiffEqOperators interface is mostly focused on nonlinear and it’s not complete right now.