A moment approach for entropy solutions to nonlinear hyperbolic PDEs

Marx Swann; Weisser Tillmann; Henrion Didier; Lasserre Jean Bernard

doi:10.3934/mcrf.2019032

Article Contents

2020, Volume 10, Issue 1: 113-140. Doi: 10.3934/mcrf.2019032

This issue Previous Article Minimal time of null controllability of two parabolic equations Next Article Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative

A moment approach for entropy solutions to nonlinear hyperbolic PDEs

1.
CNRS, LAAS, Université de Toulouse, 7 avenue du colonel Roche, F-31400 Toulouse, France
2.
Applied Mathematics and Plasma Physics Group and Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, NM 87545 Los Alamos, USA
3.
Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 4, CZ-16206 Prague, Czechia
4.
Institute of Mathematics of Toulouse (IMT), Université Paul Sabatier, 118 Route de Narbonne, F-31400, Toulouse, France

^* Corresponding author: Swann Marx
^* Corresponding author: Swann Marx

Received: July 2018

Revised: February 2019

Early access: April 2019

Published: March 2020

This work was partly funded by the ERC Advanced Grant Taming and by project 16-19526S of the Grant Agency of the Czech Republic. Part of the research of the second author was also supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project numbers 20180468ER and 20170508DR

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Abstract

We propose to solve hyperbolic partial differential equations (PDEs) with polynomial flux using a convex optimization strategy.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.

Keywords:

Nonlinear partial differential equation,
conservation laws,
numerical analysis,
convex optimization,
moments and positive polynomials.

Mathematics Subject Classification: Primary: 35L60, 35L67; Secondary: 90C26.

Citation:

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Figure 1. Approximation of the solution $ y(t,x) $ obtained with our GMP approach, in the case of a shock

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Figure 2. Approximation of the solution $ y(t,x) $ obtained with our GMP approach, in the case of a rarefaction wave

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Table 1. Approximation of $ y(0.75,x) $ with Godunov and GMP

$ x $	0.1850	0.1855	0.1860	0.1865	0.1870	0.1875	0.1880	0.1885
Godunov	0.9999	0.9991	0.9936	0.9580	0.7647	0.2724	0.0123	0.0000
GMP	1.0000	1.0000	1.0000	1.0000	1.0000	0.0000	0.0000	0.0000

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