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A moment approach for entropy solutions to nonlinear hyperbolic PDEs

  • * Corresponding author: Swann Marx

    * Corresponding author: Swann Marx 

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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  • 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.

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


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

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

    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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