Well-posedness, long-time behavior, and discretization of some models of nonlinear acoustics in velocity-enthalpy formulation

TL;DR

This paper analyzes nonlinear acoustics models using a velocity-enthalpy formulation, establishing well-posedness, long-term behavior, and structure-preserving discretizations that are validated through numerical tests.

Contribution

It introduces a velocity-enthalpy formulation for nonlinear acoustics models, enabling well-posedness analysis and structure-preserving discretizations not previously explored.

Findings

Solutions exist and are unique for small data

Solutions converge exponentially fast to equilibrium

Numerical tests confirm the effectiveness of the discretization

Abstract

We study a class of models for nonlinear acoustics, including the well-known Westervelt and Kuznetsov equations, as well as a model of Rasmussen that can be seen as a thermodynamically consistent modification of the latter. Using linearization, energy estimates, and fixed-point arguments, we establish the existence and uniqueness of solutions that, for sufficiently small data, are global in time and converge exponentially fast to equilibrium. In contrast to previous work, our analysis is based on a velocity-enthalpy formulation of the problem, whose weak form reveals the underlying port-Hamiltonian structure. Moreover, the weak form of the problem is particularly well-suited for a structure-preserving discretization. This is demonstrated in numerical tests, which also highlight typical characteristics of the models under consideration.