Finite Volume Perspectives on Finite Difference Schemes and Boundary Formulations for Wave Simulation

Time-domain finite difference (FD) and digital waveguide mesh (DWM) methods have seen extensive exploration as techniques for physical modelling sound synthesis and artificial reverberation. Various formulations of these methods have been unified under the FD framework, but many discrete boundary models important in room acoustics applications have not been. In this paper, the finite volume (FV) framework is used to unify various FD and DWM topologies, as well as associated boundary models. Additional geometric insights on existing stability conditions provide guidance into the FV meshing pre-processing step necessary for the acoustic modelling of irregular and realistic room geometries. DWM “1-D” boundary terminations are shown, through an equivalent FV formulation, to have a consistent multidimensional interpretation that is approximated to second-order accuracy, however the geometry and wall admittances being approximated may vary from what is desired. It is also shown that certain re-entrant corner configurations can lead to instabilities and an alternative stable update is provided for one problematic configuration.