Algorithmic differentiation of hyperbolic flow problems

TL;DR

This paper develops an algorithmic differentiation framework tailored for hyperbolic PDEs with shocks, enabling accurate sensitivity computations where black-box methods fail, demonstrated on Burgers and Euler equations.

Contribution

The paper introduces a novel calculus-based algorithmic differentiation method specifically designed for hyperbolic PDEs with shocks, improving sensitivity analysis accuracy.

Findings

Correct sensitivities computed for Burgers and Euler equations

Black-box AD fails for shock problems

Modified CFD code successfully incorporates shock sensitivity

Abstract

We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus introduced in Bressan and Marson [Rend. Sem. Mat. Univ. Padova, 94:79-94, 1995]. Numerical results are presented for the one-dimensional Burgers equation and the Euler equations. Using the essential routines of a state-of-the-art code for computational fluid dynamics (CFD) as a starting point, three modifications are required to apply the introduced calculus. First, the CFD code is modified to solve an additional equation for the shock location. Second, we customize the computation of the corresponding tangent to the shock location. Finally, the modified method is enhanced by algorithmic differentiation. Applying the introduced calculus to problems of the Burgers equation and the Euler equations, it is found that correct sensitivities can be computed, whereas the application of black-box algorithmic differentiation fails.