A mimetic discretization of Westervelt's equation

Abstract

A broad class of nonlinear acoustic wave models possess a Hamiltonian structure in their dissipation-free limit and a gradient flow structure for their dissipative dynamics. This structure may be exploited to design numerical methods which preserve the Hamiltonian structure in the dissipation-free limit, and which achieve the correct dissipation rate in the spatially-discrete dissipative dynamics. Moreover, by using spatial discretizations which preserve the de Rham cohomology, the non-evolving involution constraint for the vorticity may be exactly satisfied for all of time. Numerical examples are given using a mimetic finite difference spatial discretization.