Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit

TL;DR

This paper establishes robust, optimal error bounds for finite element and semi-implicit discretizations of the Kuznetsov equation, a nonlinear acoustic wave model, in the inviscid limit, supported by numerical experiments.

Contribution

It provides the first rigorous error analysis for the Kuznetsov equation's discretizations that remain valid as damping vanishes, addressing a significant gap in numerical analysis.

Findings

Optimal error bounds are achieved for the discretizations.

Error estimates are robust with respect to the vanishing damping parameter.

Numerical experiments confirm the theoretical results.

Abstract

The Kuznetsov equation is a classical wave model of acoustics that incorporates quadratic gradient nonlinearities. When its strong damping vanishes, it undergoes a singular behavior change, switching from a parabolic-like to a hyperbolic quasilinear evolution. In this work, we establish for the first time the optimal error bounds for its finite element approximation as well as a semi-implicit fully discrete approximation that are robust with respect to the vanishing damping parameter. The core of the new arguments lies in devising energy estimates directly for the error equation where one can more easily exploit the polynomial structure of the nonlinearities and compensate inverse estimates with smallness conditions on the error. Numerical experiments are included to illustrate the theoretical results.