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A mimetic discretization of Westervelt's equation

William Barham, Philip J. Morrison

TL;DR

This work develops a structure-preserving discretization for Westervelt's nonlinear acoustic equation by exploiting its Hamiltonian structure in the dissipation-free limit and a gradient-flow–driven dissipation. A general mimetic discretization framework that preserves the de Rham cohomology is combined with a Strang-splitting time integrator to ensure accurate energy dissipation and exact vorticity preservation in the discrete setting. The method, demonstrated in 1D and 2D on periodic domains, achieves expected convergence rates and faithfully reproduces both energy decay and vorticity conservation, even under spatially varying sound speed. The approach is versatile and compatible with Galerkin or collocation schemes, and it offers a pathway to structure-preserving discretizations for a broader class of nonlinear acoustic models.

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.

A mimetic discretization of Westervelt's equation

TL;DR

This work develops a structure-preserving discretization for Westervelt's nonlinear acoustic equation by exploiting its Hamiltonian structure in the dissipation-free limit and a gradient-flow–driven dissipation. A general mimetic discretization framework that preserves the de Rham cohomology is combined with a Strang-splitting time integrator to ensure accurate energy dissipation and exact vorticity preservation in the discrete setting. The method, demonstrated in 1D and 2D on periodic domains, achieves expected convergence rates and faithfully reproduces both energy decay and vorticity conservation, even under spatially varying sound speed. The approach is versatile and compatible with Galerkin or collocation schemes, and it offers a pathway to structure-preserving discretizations for a broader class of nonlinear acoustic models.

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.
Paper Structure (32 sections, 4 theorems, 131 equations, 4 figures, 2 tables)

This paper contains 32 sections, 4 theorems, 131 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The continuous Hamiltonian theory of nonlinear acoustics
  3. Hamiltonian field theories and dissipative dynamics
  4. General nonlinear acoustic models
  5. An abstract formulation of the Hamiltonian structure of nonlinear acoustics
  6. Westervelt's equation and its Hamiltonian structure
  7. A discrete Hamiltonian theory of nonlinear acoustics
  8. A general framework for mimetic discretization
  9. Discrete duality pairings and discrete functional calculus
  10. A mimetic discretization of nonlinear acoustics
  11. A Hamiltonian structure-preserving discretization of Westervelt's equation
  12. A mimetic discretization of Westervelt's equation
  13. Structure-preserving time integration
  14. Partial flow for the conservative dynamics
  15. Partial flow for the dissipative dynamics
  16. ...and 17 more sections

Key Result

Theorem 1

Let $K: V \to \mathbb{R}$ be an arbitrary functional and let $\mathsf{K} := K \circ \mathcal{I}: \mathcal{C} \to \mathbb{R}$ represent the discrete analog of the functional $K$. Moreover, define $\boldsymbol{\mathsf{u}} = \mathcal{R} u$ and $\boldsymbol{\mathsf{v}} = \mathcal{R} v$. Then Therefore, $D(K \circ \Pi): V \to \mathbb{R}$ is defined by the map

Figures (4)

  • Figure 1: The evolution of a Gaussian wave-form predicted by the one-dimensional solver.
  • Figure 2: A comparison of the actual and predicted energy dissipation rate from the one-dimensional solver.
  • Figure 3: Visualization of the solution to Westervelt's equation in two dimensions with spatially varying sound speed.
  • Figure 4: Scalar diagnostics for two-dimensional solver. Top: relative error between the actual and predicted energy dissipation rate. Bottom: relative error in the vorticity field as a function of time.

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3