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

David I. Ketcheson, Abhijit Biswas

TL;DR

The paper tackles the problem of approximating arbitrarily high-order evolution PDEs by first-order hyperbolic relaxation, enabling causal, efficiently solvable reformulations. It develops a general construction that introduces auxiliary variables and uses a signed permutation to enforce consistency between the auxiliary constraints, proving a unique stable hyperbolization for linear scalar high-order PDEs and establishing convergence to the original equation as the relaxation time $\tau \to 0$. The authors prove a Fourier-space convergence result with an $O(\tau)$ error, and extend the approach to nonlinear equations including the nonlinear Schrödinger equation, Camassa–Holm, and Kuramoto–Sivashinsky, demonstrating numerical convergence in representative examples. This work broadens the applicability of hyperbolic relaxation, offering a unified methodology to transform complex high-order models into tractable, causally consistent hyperbolic systems with potential computational and analytical 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.

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

TL;DR

The paper tackles the problem of approximating arbitrarily high-order evolution PDEs by first-order hyperbolic relaxation, enabling causal, efficiently solvable reformulations. It develops a general construction that introduces auxiliary variables and uses a signed permutation to enforce consistency between the auxiliary constraints, proving a unique stable hyperbolization for linear scalar high-order PDEs and establishing convergence to the original equation as the relaxation time . The authors prove a Fourier-space convergence result with an error, and extend the approach to nonlinear equations including the nonlinear Schrödinger equation, Camassa–Holm, and Kuramoto–Sivashinsky, demonstrating numerical convergence in representative examples. This work broadens the applicability of hyperbolic relaxation, offering a unified methodology to transform complex high-order models into tractable, causally consistent hyperbolic systems with potential computational and analytical 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.
Paper Structure (18 sections, 6 theorems, 62 equations, 6 figures)

This paper contains 18 sections, 6 theorems, 62 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Prior work and motivation
  3. The Heat Equation
  4. Relation to Jin-Xin Relaxation
  5. The Korteweg-de Vries Equation
  6. A general approach to hyperbolization
  7. Necessary Conditions for Stability
  8. Low-wavenumber stability
  9. High-wavenumber stability
  10. The unique stable hyperbolization
  11. General Linear Scalar Evolution PDEs
  12. Accuracy of the hyperbolic approximation
  13. Proof of Theorem \ref{['thm:convergence']}
  14. Additional Nonlinear Examples
  15. The Nonlinear Schrodinger Equation
  16. ...and 3 more sections

Key Result

Lemma 1

Let $P$ be a real signed permutation matrix with all eigenvalues in the closed left half-plane. Then where $R$ is skew-symmetric ($R^T = -R$), and $D$ is a diagonal matrix with all entries equal to 1 or 0.

Figures (6)

  • Figure 1: Comparison of the solution of the heat equation \ref{['heat']} and its hyperbolic approximation \ref{['heatH']}. The approximation improves with smaller values of $\tau$.
  • Figure 2: Comparison of the solution of the KdV equation \ref{['kdv']} and its hyperbolic approximation \ref{['kdvH']}. The approximation improves with smaller values of $\tau$.
  • Figure 3: Comparison of the solution of the NLS equation \ref{['nls']} and its hyperbolic approximation \ref{['nlsH']}. The approximation improves with smaller values of $\tau$.
  • Figure 4: Comparison of solution of the Cammasa-Holm equation and its hyperbolic approximation. The approximation improves with smaller values of $\tau$.
  • Figure 5: Solution of the KS equation \ref{['KS']} up to $t=50$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 3
  • ...and 2 more