Finite difference scheme for 2D sound propagation

Question

I am simulating the sound wave propagation in non-rectangular and asymetric spaces using finite-difference method. I presume linear acoustics equations to be enough (i.e. $\Box p = 0$, $\Box \vec{v} = 0$) and I need to have open and closed boundaries of the system (e.g. node or antinode of acoustic pressure on the wall and the same for velocity).

The problem occurs when I have to deal with the Neumann boundary conditions. The method is unstable even for dense and computationally inconvenient grids.

The physics behind the maths allows 3+1 more or less straightforward possible ways how to do the simulation:

• Model the acoustic pressure (then the Neumann condition is needed at the open/loose boundaries: $\partial_x p = 0$ on the boundary, closed boundaries are OK)
• Model the velocity potential $\phi$ (not very convenient cause $\partial_x \phi = 0$ on the open boundary and $\partial_t \phi = 0$ on the closed boundary)
• Model the acoustic velocity (i.e. model d'Alembert wave equation for the vector quantity - not very elegant, that's why we have potential)
• Reformulate the physics to be e.g. hamiltonian (i.e. have only the first order derivatives), lattice Boltzmann etc. etc.

Any suggestions or experiences please?

Edit:

CFL condition is satisfied.

The problem actually might be the numerical dispersion. The picture shows a system with the source on the left oscillating from -1 to 1 sinusoidally. The values on the picture outside the source area vary between -0.1 and 0.1. On the top and bottom there are Dirichlet boundaries, there is Neumann boundary at the right.

FDM with centered time and space was used. In matlab (dx=dy):

A(ii,jj,n+1)=2*A(ii,jj,n)-A(ii,jj,n-1)+c1^2*dT^2/dX^2*(A(ii-1,jj,n)+A(ii+1,jj,n)+A(ii,jj-1,n)+A(ii,jj+1,n)-4*A(ii,jj,n));

Answer 1

For numerical simulations of hyperbolic equations finite volume methods have been used in the literature.

I found clawpack and the book Finite-Volume Methods for Hyperbolic Problems from its author Randall J. LeVeque very helpful for implementing my own code. Since finite volume and finite differences are not so different, changing to the finite volume code might be a way to solve this.

With finite differences one has either problems with numerical dispersion or with numerical diffusion. To counter these different flux limiter functions are used to handle this short coming in finite volume codes.

You might also just implement your system with clawpacks python api, for reference, check out Gallery acoustics.