Numerische Mathematik· 2026Q1
Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit
- 2citations
- Q1SCImago
- 2026year
Short summary
A new analytical framework establishes optimal error bounds for finite element and semi-implicit fully discrete approximations of the Kuznetsov equation, robustly handling the vanishing damping parameter.
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Key points
- Devised an analytical framework for optimal error bounds of Kuznetsov equation approximations.
- Established robust error bounds for finite element and semi-implicit fully discrete methods.
- The bounds are effective even as the damping parameter vanishes, a regime with singular behavior change.
- The method uses energy estimates on the error equation, exploiting polynomial nonlinearities and small error conditions.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract The Kuznetsov equation is a classical wave model of nonlinear 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 devise an analytical framework that allows establishing 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 deriving suitable 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.
The authors' abstract, as published at the source. Numerische Mathematik, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics