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

Jan Giesselmann; Hendrik Ranocha

doi:10.5802/jcm.144

¹Department of Mathematics, Technical University of Darmstadt, Germany
²Institute of Mathematics, Johannes Gutenberg University Mainz, Germany

The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 75-102

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.

Published online: 2026-03-02

DOI: 10.5802/jcm.144

Classification: 35L03, 35M11, 65M12, 65M06, 65M20
Keywords: hyperbolization, hyperbolic relaxation, relative energy, relative entropy, structure-preserving methods, energy-conserving methods, summation-by-parts operators, Benjamin–Bona–Mahony equation, Korteweg–de Vries equation, Gardner equation, Kawahara equation, Kuramoto–Sivashinsky equation

Author's affiliations:

Jan Giesselmann ¹ ; Hendrik Ranocha ²

¹ Department of Mathematics, Technical University of Darmstadt, Germany
² Institute of Mathematics, Johannes Gutenberg University Mainz, Germany

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Jan Giesselmann; Hendrik Ranocha. Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 75-102. doi: 10.5802/jcm.144

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