An explicit symplectic approach to solving the wave equation in moving media

An explicit approach using symplectic time integration in conjunction with traditional finite difference spatial derivatives to solve the wave equation in moving media is presented. A simple operator split of this second order wave equation into two coupled first order equations is performed, allowing these split equations to be solved symplectically. Orders of symplectic time integration ranging from first to fourth along with orders of spatial derivatives ranging from second to sixth are explored. The case of cylindrical acoustic spreading in air under a constant velocity in a 2D square structured domain is considered. The variation of the computed time-of-flight, frequency, and wave length are studied with varying grid resolution and the deviations from the analytical solutions are determined. It was found that symplectic time integration interferes with finite difference spatial derivatives higher than second order causing unexpected results. This is actually beneficial for unstructured finite volume tools like OpenFOAM where second order spatial operators are the state-of-the art. Cylindrical acoustic spreading is simulated on an unstructured 2D triangle mesh showing that symplectic time integration is not limited to the spatial discretization paradigm and overcomes the numerical diffusion arising with the in-built numerical methods which hinder wave propagation.