A mimetic discretization of Westervelt's equation

TL;DR

This paper introduces a mimetic finite difference discretization for Westervelt's nonlinear acoustic wave equation, preserving key geometric structures and constraints to improve numerical accuracy and physical fidelity.

Contribution

It develops a structure-preserving discretization method that maintains Hamiltonian and cohomological properties for nonlinear acoustic models.

Findings

Preserves Hamiltonian structure in the dissipation-free limit

Exactly satisfies the vorticity involution constraint

Demonstrates improved numerical accuracy with mimetic discretization

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.