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.

翻译：我们提出了一种构建一阶双曲系统的框架，其解可逼近目标高阶演化方程的解。近年来，此类构造方法受到越来越多的关注，并可能作为分析或计算工具，用于理解相应的高阶方程。我们对一族线性模型方程进行了系统分析，结果表明：对于该族中的每个方程，均存在一个稳定的双曲逼近，其解在特定极限下收敛于模型方程的解。随后，我们通过若干实例证明，该方法可成功应用于众多具有实际意义的非线性偏微分方程。