Automatic differentiation for performing the Cauchy-Kovalevskaya procedure in Lax-Wendroff type discretizations

TL;DR

This paper introduces an automatic differentiation approach for the Cauchy-Kovalevskaya procedure in Lax-Wendroff methods, enabling high-order, positivity-preserving, and efficient flux computations for hyperbolic conservation laws.

Contribution

It presents a Jacobian-free, problem-independent AD method that simplifies high-order flux calculations without positivity corrections, outperforming traditional approximate methods.

Findings

Demonstrates order and positivity preservation in numerical experiments.

Shows AD method's wall-clock time is comparable to approximate Lax-Wendroff.

Applicable to any physical flux function and method order.

Abstract

Lax-Wendroff methods combined with discontinuous Galerkin/flux reconstruction spatial discretization provide a high-order, single-stage, quadrature-free method for solving hyperbolic conservation laws. In this work, we introduce automatic differentiation (AD) for performing the Cauchy-Kowalewski procedure used in the element-local time average flux computation step (the predictor step) of Lax-Wendroff methods. The application of AD is similar for methods of any order and does not need positivity corrections during the predictor step. This contrasts with the approximate Lax-Wendroff procedure, which requires different finite difference formulas for different orders of the method and positivity corrections in the predictor step for fluxes that can only be computed on admissible states. The method is Jacobian-free and problem-independent, allowing direct application to any physical flux function. Numerical experiments demonstrate the order and positivity preservation of the method. Additionally, performance comparisons indicate that the wall-clock time of automatic differentiation is always on par with the approximate Lax-Wendroff method.