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

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.