Absorbing boundary conditions for the Westervelt equation

TL;DR

This paper develops high-order nonlinear absorbing boundary conditions for the Westervelt equation using pseudo-differential calculus, improving accuracy and efficiency in simulating wave propagation in one and two dimensions.

Contribution

It introduces a novel family of high-order nonlinear absorbing boundary conditions for the Westervelt equation, constructed via pseudo-differential calculus, with proven local well-posedness and demonstrated numerical efficiency.

Findings

High-order boundary conditions outperform low-order ones in accuracy.

Numerical experiments confirm the effectiveness of the boundary conditions.

The approach is applicable in both one and two spatial dimensions.

Abstract

The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation.