Approximation of arbitrarily high-order PDEs by first-order hyperbolic relaxation

TL;DR

This paper introduces a systematic framework for approximating high-order PDEs with first-order hyperbolic systems, enabling better analytical and computational handling of complex evolution equations.

Contribution

The authors develop a general method for constructing stable hyperbolic approximations of high-order PDEs, with convergence guarantees and broad applicability to nonlinear equations.

Findings

Stable hyperbolic approximations converge to high-order PDE solutions.

Method applies successfully to various nonlinear PDEs.

Framework provides analytical and computational advantages.

Abstract

We present a framework for constructing a first-order hyperbolic system whose solution approximates that of a desired higher-order evolution equation. Constructions of this kind have received increasing interest in recent years, and are potentially useful as either analytical or computational tools for understanding the corresponding higher-order equation. We perform a systematic analysis of a family of linear model equations and show that for each member of this family there is a stable hyperbolic approximation whose solution converges to that of the model equation in a certain limit. We then show through several examples that this approach can be applied successfully to a very wide range of nonlinear PDEs of practical interest.