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An Approximate Lax-Wendroff-Type Procedure for High Order Accurate Schemes for Hyperbolic Conservation Laws

David Zorío, Antonio Baeza, Pep Mulet

TL;DR

This work develops an approximate Lax–Wendroff‑type procedure for high‑order time integration of hyperbolic conservation laws, replacing expensive exact flux‑derivative expressions with high‑order central differences. The method preserves conservation and extends naturally to multiple dimensions and systems, with a proven $R$‑th order accuracy and a conservative flux form. Numerical experiments across 1D and 2D problems (including Euler flows) show the approximate method delivers results essentially indistinguishable from the exact approach while offering simpler implementation and improved efficiency relative to Runge–Kutta time stepping. The approach yields practical, high‑fidelity solvers for complex hyperbolic systems, maintaining accuracy at high orders with reduced symbolic complexity.

Abstract

A high order time stepping applied to spatial discretizations provided by the method of lines for hyperbolic conservations laws is presented. This procedure is related to the one proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185-2198, 2003) for numerically solving hyperbolic conservation laws. Both methods are based on the conversion of time derivatives to spatial derivatives through a Lax-Wendroff-type procedure, also known as Cauchy-Kovalevskaya process. The original approach in Qiu and Shu (2003) uses the exact expressions of the fluxes and their derivatives whereas the new procedure computes suitable finite difference approximations of them ensuring arbitrarily high order accuracy both in space and time as the original technique does, with a much simpler implementation and generically better performance, since only flux evaluations are required and no symbolic computations of flux derivatives are needed.

An Approximate Lax-Wendroff-Type Procedure for High Order Accurate Schemes for Hyperbolic Conservation Laws

TL;DR

This work develops an approximate Lax–Wendroff‑type procedure for high‑order time integration of hyperbolic conservation laws, replacing expensive exact flux‑derivative expressions with high‑order central differences. The method preserves conservation and extends naturally to multiple dimensions and systems, with a proven ‑th order accuracy and a conservative flux form. Numerical experiments across 1D and 2D problems (including Euler flows) show the approximate method delivers results essentially indistinguishable from the exact approach while offering simpler implementation and improved efficiency relative to Runge–Kutta time stepping. The approach yields practical, high‑fidelity solvers for complex hyperbolic systems, maintaining accuracy at high orders with reduced symbolic complexity.

Abstract

A high order time stepping applied to spatial discretizations provided by the method of lines for hyperbolic conservations laws is presented. This procedure is related to the one proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185-2198, 2003) for numerically solving hyperbolic conservation laws. Both methods are based on the conversion of time derivatives to spatial derivatives through a Lax-Wendroff-type procedure, also known as Cauchy-Kovalevskaya process. The original approach in Qiu and Shu (2003) uses the exact expressions of the fluxes and their derivatives whereas the new procedure computes suitable finite difference approximations of them ensuring arbitrarily high order accuracy both in space and time as the original technique does, with a much simpler implementation and generically better performance, since only flux evaluations are required and no symbolic computations of flux derivatives are needed.

Paper Structure

This paper contains 21 sections, 5 theorems, 109 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Original Lax-Wendroff-type procedure
  3. The approximate Lax-Wendroff-type procedure
  4. Scheme formulation
  5. Fifth order scheme example
  6. Theoretical results
  7. Numerical experiments
  8. 1D Linear advection equation
  9. 1D Burgers equation
  10. Smooth solution
  11. Discontinuous solution
  12. 1D Euler equations
  13. Smooth solution
  14. Shu-Osher problem
  15. Blast wave
  16. ...and 6 more sections

Key Result

theorem 1

Let $f:\mathbb{R}^m\rightarrow\mathbb{R}^{p},$$u:\mathbb{R}\rightarrow\mathbb{R}^m$$n$ times continuously differentiable. Then where $\mathcal{P}_{n}=\{ s\in\mathbb N^{n} / \sum_{j=1}^{n} j s_j=n \}$, $|s|=\sum_{j=1}^{n} s_j$, $\left(\right)=\frac{n!}{s_1!\cdots s_n!}$, $D^s u(t)$ is an $m\times |s|$ matrix whose ($\sum\limits_{l<j}s_l+i$)-th column is given by and the action of the $k$-th deriv

Figures (4)

  • Figure 1: Discontinuous solution for Burgers equation, $t=12$.
  • Figure 2: Shu-Osher problem. Pressure field.
  • Figure 3: Blast wave.
  • Figure 4: Double Mach Reflection. Pressure field.

Theorems & Definitions (10)

  • theorem 1: Faà di Bruno's formula
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proof
  • theorem 2
  • proof
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • proof