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

TL;DR

This work addresses achieving high-order one-step time discretization for hyperbolic conservation laws by integrating automatic differentiation (AD) into the Cauchy-Kovalevskaya predictor of Lax-Wendroff flux reconstruction (LWFR). Unlike the traditional approximate LW procedure that relies on order-specific finite-difference derivatives and can require positivity corrections, the AD-based approach computes temporal flux derivatives $\partial_t^m \boldsymbol{f}$ in a Jacobian-free, problem-independent manner, applicable to any flux function, and supports orders of accuracy with a uniform framework. The authors implement both point-wise and element-wise AD variants and demonstrate that the AD predictor preserves admissibility and outperforms the approximate LW method in wall-clock time across multiple 1D and 2D test problems, including isentropic Euler and relativistic hydrodynamics. The results show that AD yields robust, high-order solutions with reduced limiting and positivity concerns, offering a practical route to scalable, high-fidelity LW-type schemes. The approach has broad applicability to DW/DG-type discretizations and can be extended to other equations, flux functions, and potential MOOD-style adaptations in troubled regions.

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.

Figures (3)

• Figure 1: Challenging Riemann problems for conservation laws whose fluxes can only be computed on admissible states solved using the LW-AD scheme with polynomial degree $N=3$ (LW3). (a) Double rarefaction Riemann problem for isentropic Euler equations at $t=0.2$, (b) Riemann problem for relativistic hydrodynamics from ryu2006equation, numbered first in basak2025bound.
• Figure 2: Results with compressible Euler's equations obtained using the LW-AD scheme with different polynomial degrees $N$ (LW-$N$) and the multiderivative Runge-Kutta flux reconstruction scheme using AD using polynomial degree $N=3$ (MDRK3). (a) Convergence plot of isentropic vortex, (b) Density profile of Mach 2000 astrophysical jet.
• Figure 3: Wall-clock time taken to compute high order derivatives of $\sin x$ at $x=1.0$ using ForwardDiff.jl, TaylorDiff.jl, and Enzyme.jl (a) Intel CPU, (b) Apple M3 CPU.

Theorems & Definitions (2)

• definition 2.1
• definition 2.2