On hyperbolic approximations for a class of dispersive and diffusive-dispersive equations

TL;DR

This paper introduces hyperbolic approximation systems for dispersive and diffusive-dispersive equations, establishing well-posedness, entropy structures, and convergence of solutions, enabling effective numerical simulations of complex nonlinear PDEs.

Contribution

The authors develop novel hyperbolic approximations for dispersive and diffusive-dispersive equations, including entropy structures and convergence proofs, facilitating numerical analysis of these complex systems.

Findings

Constructed a first-order hyperbolic approximation for dispersive equations.

Proved convergence of approximate solutions to original equations.

Validated the approach through numerical tests on various nonlinear equations.

Abstract

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth solutions is achieved by constructing a unique symmetrizer that applies to arbitrary smooth fluxes. Under stronger conditions on the fluxes, we provide a strictly convex entropy for the hyperbolic system that corresponds to the energy of the underlying dispersive equation. To approximate diffusive-dispersive equations, we rely on a viscoelastic damped system that is compatible with the found entropy for the hyperbolic approximation of the dispersive evolution. For the resulting hyperbolic-parabolic approximation, we provide a global well-posedness result. Using the relative entropy framework \cite{dafermos2005hyperbolic}, we prove that the solutions of the approximate systems converge to solutions of the original equations. The structure of the new approximate systems allows to apply standard numerical simulation methods from the field of hyperbolic balance laws. We confirm the convergence of our approximations even beyond the validity range of our theoretical findings on set of test cases covering different target equations. We show the applicability of the approach for strong nonlinear effects leading to oscillating or shock-layer-forming behavior.