A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

TL;DR

This paper introduces a moment-based numerical method for solving parameter-dependent hyperbolic PDEs using a hierarchy of convex optimization problems, enabling accurate solution reconstruction and uncertainty quantification.

Contribution

It develops a novel approach combining parametric entropy measure-valued solutions with semidefinite programming to approximate solutions of hyperbolic PDEs with convergence guarantees.

Findings

Convergent sequence of moment approximations for entropy MV solutions.

Ability to reconstruct solution graphs using Christoffel-Darboux kernels.

Effective uncertainty quantification for parametrized PDEs.

Abstract

We propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel-Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.