A moment approach for entropy solutions to nonlinear hyperbolic PDEs

TL;DR

This paper introduces a novel convex optimization-based method for solving nonlinear hyperbolic PDEs using measure-valued solutions, providing a hierarchy of finite-dimensional problems and demonstrating its effectiveness on Burgers' equation.

Contribution

It develops a new approach that approximates measure-valued solutions of hyperbolic PDEs through convex optimization and establishes conditions for their equivalence.

Findings

Method successfully applied to Burgers' equation

Hierarchy of convex problems approximates solutions effectively

Comparison shows advantages over traditional schemes

Abstract

We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is then illustrated on the celebrated Burgers equation. We also compare our results with an existing numerical scheme, namely the Godunov scheme.