Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping

TL;DR

This work develops and analyzes a time-stepping scheme for the nonlinear Westervelt equation with nonlocal in time damping, modeled by kernels $\beta_{\text{A}}$ and $\beta_{\text{B}}$ and implemented via trapezoidal time stepping with convolution quadrature. By combining a rigorous well-posedness framework, energy estimates, and a fixed-point argument, the authors establish stability and local existence for the nonlinear integro-differential problem, and derive sharp error bounds for the semi-discrete scheme. A correction to the convolution quadrature is shown to be essential for achieving the higher-order convergence suggested by the fractional damping, yielding $O(\Delta t^{1+\mu})$ accuracy. Numerical experiments in 1D and 2D corroborate the theory, demonstrating expected convergence rates and highlighting the influence of nonlinear, memory, and damping parameters on wave propagation phenomena in nonlinear acoustics.

Abstract

We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

Figures (10)

• Figure 1: Convergence of the maximum energy error for the numerical scheme without the correction term for $\mu = 0.25$ and $\mu = 0.75$. Predicted convergence order of $\mathcal{O}({\Delta t})$ is also shown.
• Figure 2: Convergence of the maximum energy error for the numerical scheme with the correction term included for $\mu = 0.25$ and $\mu = 0.75$. Predicted convergence order of $\mathcal{O}({\Delta t}^{1+\mu})$ is also shown.
• Figure 3: Solution of \ref{['eq:1']} at $T=4$ approximated with the scheme \ref{['equ:full_disc']} with $a=0$ for various values of $k$.
• Figure 4: Solution of \ref{['eq:1']} at $T=4$ approximated using the scheme \ref{['equ:full_disc']} with $a=1$, $\beta =\beta_{\text{A}}$, $\mu = 0.5$, and $r=0$, for various values of $k$.
• Figure 5: Solution of \ref{['eq:1']} at various time points up to $T=4$ approximated with the scheme \ref{['equ:full_disc']} with $k=0.09$, $\mu = 0.5$, and $r=0$, for various values of constant $a$.

Theorems & Definitions (19)

• Lemma 2.1
• proof
• Remark 2.2
• Definition 2.3
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Theorem 3.4