Automatic differentiation and its application to Lax-Wendroff methods

Abstract:   Automatic differentiation (AD) is a family of techniques for computing derivatives of functions using a computer program. It works by systematically applying the chain rule, unlike symbolic differentiation which manipulates algebraic expressions, or numerical approximations which rely on finite differences. AD has wide applications in scientific computing, including optimization, sensitivity analysis, and machine learning. This talk will begin with a simple demonstration of how dual numbers enable AD, followed by the application of AD to Lax-Wendroff type discretizations.

Lax–Wendroff methods combined with discontinuous Galerkin or flux reconstruction spatial discretizations, provide high-order, single-stage, quadrature-free schemes for hyperbolic conservation laws. In these methods, we propose the application of AD in the element-local time-average flux computation (the predictor step). 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.

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