Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems

Guermond, Jean‐Luc; Popov, Bojan

doi:10.1137/16m1074291

Cited by 112 publications

(197 citation statements)

References 28 publications

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“…Entropy stability [8,29,33,34] and preservation of invariant domains [14,16,19] play an important role in the design of numerical methods for nonlinear hyperbolic conservation laws. A failure to comply with these design criteria may result in nonphysical artefacts and/or convergence to wrong weak solutions.…”

Section: Introductionmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

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In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable / entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

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Section: Introductionmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

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“…This scheme satisfies local invariant domain properties and local discrete entropy inequalities when the bottom is flat. It is an adaptation of the method presented in Audusse et al [2] to the continuous finite element setting developed in Guermond and Popov [16]. To the best of our knowledge, this is the first result of this type for continuous finite elements.…”

Section: Definition 22 (Centrosymmetry)mentioning

confidence: 92%

“…Reduction to the one-dimensional Riemann problem. For completeness, we show how the estimation of λ f max (n, U L , U R ) can be reduced to estimating the maximum wave speed in a one-dimensional Riemann problem independent of n. Similarly to [16], we make a change of basis and introduce t 1 , . .…”

Section: Full Time and Space Approximation Letmentioning

confidence: 99%

“…The first building block of the method consists of using the methodology introduced in Guermond and Popov [16]. The second building block consists of making the schemes well-balanced with respect to rest states by using the so-called hydrostatic reconstruction from [2, section 2.1] and variations thereof.…”

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confidence: 99%

“…The second building block consists of making the schemes well-balanced with respect to rest states by using the so-called hydrostatic reconstruction from [2, section 2.1] and variations thereof. The technique from [16] is a loose extension of Lax's scheme [24, p. 163] to continuous finite elements; it solves general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on nonuniform grids. The artificial dissipation is defined so that any convex invariant set containing the initial data is an invariant domain for the method.…”

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confidence: 99%

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Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite Elements

Azérad¹,

Guermond²,

Popov³

2017

SIAM J. Numer. Anal.

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Abstract. This paper investigates a first-order and a second-order approximation technique for the shallow water equation with topography using continuous finite elements. Both methods are explicit in time and are shown to be well-balanced. The first-order method is invariant domain preserving and satisfies local entropy inequalities when the bottom is flat. Both methods are positivity preserving. Both techniques are parameter free, work well in the presence of dry states, and can be made high order in time by using strong stability preserving time stepping algorithms. 1. Introduction. The objective of this paper is to develop an invariant domain preserving well-balanced approximation of the shallow water equation with bathymetry using continuous finite elements. There are many finite volume and discontinuous Galerkin (DG) techniques available in the literature that can solve this problem efficiently up to second and higher order in space. Examples of schemes that are well balanced at rest and robust in the presence of dry states can be found, for example, in Audusse [28]. We refer the reader to the book of Bouchut [7] for a review on this topic, to the paper of Xing and Shu [32] for a survey on finite volume and DG methods, and to the paper [23] for a survey of central-upwind schemes. However, to the best of our knowledge, these types of approximations are not developed in the context of continuous finite elements. Or we should say that no robust continuous finite element technique is yet available in the literature that guarantees second-order accuracy, works properly in every regime (subcritical, transcritical, transcritical with hydraulic jumps, wet, and dry regions) and is well-balanced at rest. We propose such a method in the present paper. Two variants of the method are discussed: one variant is first-order accurate in space, positivity preserving, and preserves every convex invariant domain of the system in the absence of bathymetry; the other variant is second-order accurate in space and positivity preserving. Both variants are explicit in time and use continuous finite elements on unstructured meshes.

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Shock capturing with the high‐order flux reconstruction method on adaptive meshes based on p4est

Xia

2022

Engineering Reports

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High order schemes have been investigated for quite a long time, and the flux reconstruction (FR) scheme proposed by Huynh recently attracts the attention of researchers due to its simplicity and efficiency. Building the framework that bridges discontinuous Galerkin (DG) and spectral difference (SD) schemes, FR recovers DG and SD conveniently with a careful selection of parameters. In this article, FR scheme is realized based on the framework of p4est, an open source adaptive mesh refinement library. The shock capturing ability of localized Laplacian artificial viscosity and in‐cell piecewise integrated solution methods are compared. Curved boundary treatment for high order schemes is adopted. The performance of developed code is estimated in both one and two dimensions including curved boundary and shock cases, and some attractive results are obtained.

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Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems

Cited by 112 publications

References 28 publications

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite Elements

Shock capturing with the high‐order flux reconstruction method on adaptive meshes based on p4est

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