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.
Giesselmann et al. (Mon,) studied this question.