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

2025/08/06 by Giesselmann, Jan, Ranocha, Hendrik · 2 citations
#35L03 #35M11 #65M06 #65M12 #65M20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2508.04112

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.

Citations

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Cited by

• Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger equations
• Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation

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