Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions

TL;DR

This paper establishes rigorous convergence results for hyperbolic approximations of various higher-order PDEs, providing a theoretical foundation and numerical validation for these methods.

Contribution

It proves convergence of hyperbolic approximations for several higher-order PDEs under smooth solutions, filling a gap in the theoretical understanding.

Findings

Convergence of hyperbolic approximations is proven for multiple higher-order PDEs.

Numerical results support the theoretical convergence analysis.

Provides a rigorous foundation for previously used approximation methods.

Abstract

We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings.