On high-accuracy L∞-stable IMEX schemes for scalar hyperbolic multi-scale equations

We present a framework to build high-accuracy IMEX schemes that fulfill the maximum principle, applied to a scalar hyperbolic multi-scale equation. Motivated by the findings in [Gottlieb, Shu, Tadmor, 2001] that implicit R-K schemes are not L∞-stable, our scheme, for which we can prove the L ∞ stability, is based on a convex combination between a first-order and a class of second-order IMEX schemes. We numerically demonstrate the advantages of our scheme, especially for discontinuous problems, and give a MOOD procedure to increase the accuracy

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior -- an approximation of the steady solution -- which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions

Développement de schémas équilibre d'ordre élevé pour des écoulements géophysiques

This manuscript is devoted to a relevant numerical approximation of the shallow-water equations with the source terms of topography and Manning friction.The first chapter concerns the study of the shallow-water equations, equipped with the aforementioned source terms. Algebraic properties of this system are first obtained. Then, we focus on its steady state solutions for the individual source terms of topography and friction.The second chapter introduces the finite volume method, which is used throughout the manuscript. One-dimensional and two-dimensional systems of conservation laws are studied, and a high-order strategy is presented.The third chapter deals with the numerical approximation of the shallow-water equations with topography and friction. We derive a scheme that preserves all the steady states; preserves the non-negativity of the water height; is able to deal with transitions between wet and dry areas. Relevant numerical experiments are presented to exhibit these properties.The fourth chapter is dedicated to extensions of the scheme developed in the third chapter. Namely, the scheme is extended to two space dimensions, and we suggest a high-order extension. Numerical experiments are once again provided, including real-world simulations.L'objectif de ce travail est de proposer un schéma numérique pertinent pour les équations de Saint-Venant avec termes source de topographie et de friction de Manning.Le premier chapitre est dédié à l'étude du système de Saint-Venant muni des termes source. Dans un premier temps, les propriétés algébriques de ce système sont obtenues. Dans un second temps, nous nous intéressons à ses états stationnaires, qui sont étudiés pour les termes source individuels de topographie et de friction.Le deuxième chapitre permet de rappeler des notions concernant la méthode des volumes finis. Nous évoquons des schémas aux volumes finis pour des systèmes de lois de conservation unidimensionnels et bidimensionnels, et nous en proposons une extension permettant d'assurer un ordre élevé de précision.Le troisième chapitre concerne la dérivation d'un schéma numérique pour les équations de Saint-Venant avec topographie et friction. Ce schéma permet de préserver tous les états stationnaires ; de préserver la positivité de la hauteur d'eau ; d'approcher les transitions entre zones mouillées et zones sèches, et ce même en présence de friction. Des cas-tests mettant en lumière les propriétés du schéma sont présentés.Le quatrième chapitre permet d'étendre le schéma proposé précédemment, pour prendre en compte des géométries bidimensionnelles et pour assurer un ordre élevé de précision. Des cas-tests numériques sont aussi présentés, y compris des simulations de phénomènes réels

Développement de schémas équilibre d'ordre élevé pour des modèles d'écoulements géophysiques

L’objectif de ce travail est de proposer un schéma numérique pertinent pour les équations de Saint-Venant avec termes source de topographie et de friction de Manning. Le premier chapitre est dédié à l’étude du système de Saint- Venant muni des termes source. Dans un premier temps, les propriétés algébriques de ce système sont obtenues. Dans un second temps, nous nous intéressons à ses états stationnaires, qui sont étudiés pour les termes source individuels de topographie et de friction. Le deuxième chapitre permet de rappeler des notions concernant la méthode des volumes finis. Nous évoquons des schémas aux volumes finis pour des systèmes de lois de conservation unidimensionnels et bidimensionnels, et nous en proposons une extension permettant d’assurer un ordre élevé de précision. Le troisième chapitre concerne la dérivation d’un schéma numérique pour les équations de Saint-Venant avec topographie et friction. Ce schéma permet : de préserver tous les états stationnaires ; de préserver la positivité de la hauteur d’eau ; d’approcher les transitions entre zones mouillées et zones sèches, et ce même en présence de friction. Des cas-tests mettant en lumière les propriétés du schéma sont présentés. Le quatrième chapitre permet d’étendre le schéma proposé précédemment, pour prendre en compte des géométries bidimensionnelles et pour assurer un ordre élevé de précision. Des cas-tests numériques sont aussi présentés, y compris des simulations de phénomènes réels.This manuscript is devoted to a relevant numerical approximation of the shallow-water equations with the source terms of topography and Manning friction. The first chapter concerns the study of the shallow-water equations, equipped with the aforementioned source terms. Algebraic properties of this system are first obtained. Then, we focus on its steady state solutions for the individual source terms of topography and friction. The second chapter introduces the finite volume method, which is used throughout the manuscript. One-dimensional and two-dimensional systems of conservation laws are studied, and a high-order strategy is presented. The third chapter deals with the numerical approximation of the shallow-water equations with topography and friction. We derive a scheme that: • preserves all the steady states; • preserves the non-negativity of the water height; • is able to deal with transitions between wet and dry areas. Relevant numerical experiments are presented to exhibit these properties. The fourth chapter is dedicated to extensions of the scheme developed in the third chapter. Namely, the scheme is extended to two space dimensions, and we suggest a highorder extension. Numerical experiments are once again provided, including real-world simulations

TVD-MOOD schemes based on implicit explicit time integration

International audienceThe context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. [18] gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its L1 stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scaleequation and numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations

On high-precision L^∞-stable IMEX schemes for scalar hyperbolic multi-scale equations

International audienceWe present a framework to build high-precision IMEX schemes that fulfill the maximum principle, applied to a scalar hyperbolic multi-scale equation. Motivated by the findings in [5] that implicit R-K schemes are not L∞ stable, our scheme, for which we can prove the L∞ stability, is based on a convex combination between a first-and a class of second-order IMEX schemes. We numerically demonstrate the advantages of our scheme, especially for discontinuous problems, and give a MOOD procedure to increase the precision

A fully well-balanced hydrodynamic reconstruction

The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a linear well-balanced high-order procedure. Several numerical experiments demonstrate the effectiveness of the numerical scheme

A simple fully well-balanced and entropy preserving scheme for the shallow-water equations

International audienceIn this communication, we consider a numerical scheme for the shallow-water system. The scheme under consideration has been proven to preserve the positivity of the water height and to be fully well-balanced, i.e. to exactly preserve the smooth moving steady state solutions of the shallow-water equations with the topography source term. The goal of this work is to prove a discrete entropy inequality satisfied by this scheme.Dans cette communication, nous nous intéressons à un schéma numérique pour le système de Saint-Venant. Nous avons prouvé que ce schéma préservait la positivité de la hauteur de l'eau et qu'il était complètement équilibre, c'est-à-dire qu'il préservait exactement les solutions stationnaires régulières à vitesse non nulle des équations de Saint-Venant avec terme source de topographie. Le but de ce travail est de prouver une inégalité d'entropie discrète satisfaite par ce schéma

TVD-MOOD schemes based on implicit explicit time integration

International audienceThe context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. [18] gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its L1 stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scaleequation and numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations

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