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

We propose a numerical method to solve parameter-dependent scalar 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 measur...

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Veröffentlicht in:	Numerische Mathematik Jg. 156; H. 4; S. 1289 - 1324
Hauptverfasser:	Cardoen, Clément, Marx, Swann, Nouy, Anthony, Seguin, Nicolas
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2024 Springer Nature B.V Springer Verlag
Schlagworte:	Approximation Burgers equation Conservation laws Entropy Functionals Initial conditions Linear equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Nonlinear equations Numerical Analysis Numerical and Computational Physics Numerical methods Parameterization Parameters Partial differential equations Semidefinite programming Simulation Theoretical 35D99 35L65 65D15 Analysis of PDEs Numerical analysis
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Abstract	We propose a numerical method to solve parameter-dependent scalar 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.
AbstractList	We propose a numerical method to solve parameter-dependent scalar 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. 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.
Author	Seguin, Nicolas Cardoen, Clément Marx, Swann Nouy, Anthony
Author_xml	– sequence: 1 givenname: Clément surname: Cardoen fullname: Cardoen, Clément email: clement.cardoen@ec-nantes.fr organization: Nantes Université, Centrale Nantes, Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629 – sequence: 2 givenname: Swann surname: Marx fullname: Marx, Swann organization: Nantes Université, Centrale Nantes, LS2N, CNRS UMR 6004 – sequence: 3 givenname: Anthony surname: Nouy fullname: Nouy, Anthony organization: Nantes Université, Centrale Nantes, Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629 – sequence: 4 givenname: Nicolas surname: Seguin fullname: Seguin, Nicolas organization: Inria, Centre de l’Université Côte d’Azur, antenne de Montpellier, Imag, UMR CNRS 5149, Université de Montpellier
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Snippet	We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous... We propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work...
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SubjectTerms	Approximation Burgers equation Conservation laws Entropy Functionals Initial conditions Linear equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Nonlinear equations Numerical Analysis Numerical and Computational Physics Numerical methods Parameterization Parameters Partial differential equations Semidefinite programming Simulation Theoretical
Title	A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws
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