Automatic differentiation for performing the Cauchy-Kovalevskaya procedure in Lax-Wendroff type discretizations
This work simplifies the implementation of high-order Lax-Wendroff methods for computational scientists solving hyperbolic conservation laws, but the improvement is incremental as it matches existing performance.
The paper introduces automatic differentiation to perform the Cauchy-Kovalevskaya procedure in Lax-Wendroff type discretizations for hyperbolic conservation laws, achieving order and positivity preservation with wall-clock time on par with the approximate method.
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.