A Class of Lagrangian–Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms

Abreu, Eduardo; Matos, Vitor Luiz de; Pérez, John; Rodríguez-Bermúdez, Panters

doi:10.1007/s10915-020-01392-w

Cited by 17 publications

(27 citation statements)

References 55 publications

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“…The proposed weak asymptotic analysis in this work, together with the Lagrangian-Eulerian approach [3,4,5,6], fits in properly with the classical theory while optimizing the mathematical computations for the construction of new accurate numerical schemes, given by Proposition 1, where their convergence, existence, and stability are established. The solution to the proposed scheme is obtained as a family of functions, {u(•,t, ε)} ε , bounded in L 1 (S 1 ) uniformly in parameter ε of the model (1.1)).…”

Section: Introductionmentioning

confidence: 75%

“…For the sake of clarity and completeness, we sketch in what follows the main three steps of the Lagrangian-Eulerian approach, namely, the construction of the set of ODE modeling the no-flow curves (Section 2.1), the Lagrangian-Eulerian conservation relations (Section 2.2) and the final projection step by piecewise linear reconstruction (Section 2.3), along with an evolution algorithm of the fully discrete nonstaggered Lagrangian-Euclidean method. As in the Lagrangian-Eulerian schemes present in [4,5,3,6], local conservation is obtained by integrating the conservation law over the region in the space-time domain where the conservation of mass flux takes place.…”

Section: Lagrangian-eulerian Schemementioning

confidence: 99%

“…Here curves σ n j± 1 2 (t) are not straight lines in general but rather solutions to the set of local nonlinear differential equations (as in [6,5] for systems):…”

Section: No-flow Curvesmentioning

confidence: 99%

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Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

Abreu¹,

Santo²,

Lambert³

et al. 2021

Preprint

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This work presents a suitable mathematical analysis to understand the properties of convergence and bounded variation of a new fully discrete locally conservative and explicit Lagrangian-Eulerian numerical scheme to the entropy solution in the sense of Kruzhkov via weak asymptotic method. We also make use of the weak asymptotic method to connect the theoretical developments with the computational approach within the practical framework of a solid numerical analysis. This method also serves to address the issue of notions of solutions, and its resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions in intricate applications. The weak asymptotic solutions we compute in this study with our novel Lagrangian-Eulerian framework are shown to coincide with classical solutions and Kruzhkov entropy solutions in the scalar case. Moreover, we present and discuss significant computational aspects by means of numerical experiments related to nontrivial problems: a nonlocal traffic model, the 2 × 2 symmetric Keyfitz-Kranzer system, and numerical studies via Wasserstein distance to explain shock interaction with the fundamental inviscid Burgers' model for fluids. Therefore, the proposed weak asymptotic analysis, when applied to the Lagrangian-Eulerian framework, fits in properly with the classical theory while optimizing the mathematical computations for the construction of new accurate numerical schemes.

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Section: Introductionmentioning

confidence: 75%

Section: Lagrangian-eulerian Schemementioning

confidence: 99%

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Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

Abreu¹,

Santo²,

Lambert³

et al. 2021

Preprint

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“…Nesta seção apresentaremos os resultados numéricos obtidos para as leis de conservação linear e de Burgers, e para o sistema de leis de balanço Shallow Water. A fim de verificar estes resultados, para o caso das leis de conservação estes foram comparadas qualitativamente com os obtidos pelo método Lagrangian-Eulerian (LEH), desenvolvido por Perez e Abreu [1,2,8]. Para o caso das equações SW, estes foram comparadas qualitativamente com os obtidos pelo Xing e Zhang em [10].…”

Section: Resultados E Experimentos Numéricosunclassified

“…As equações que expressam a conservação e balanceamento de uma quantidade física são conhecidas como leis de conservação e balanço, respectivamente. Dentre as muitas leis de conservação e balanço conhecidas podemos citar as equações de advecção linear, de Burgers e o sistema shallow water (SW) [1,6,7,9].…”

Section: Introductionunclassified

Aplicação de esquemas Galerkin Descontínuo em problemas hiperbólicos em 2D

Araujo¹,

Bermúdez²,

Abreu³

et al. 2021

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Neste trabalho utilizamos o método de elementos finitos Galerkin Descontinuo Runge-Kutta (RKDG) em uma malha triangular para obter soluções numéricas para problemas hiperbólicos em 2D, em particular para leis de conservação escalares e para o sistema de leis de balanço shallow water. Este método consiste na discretização espacial através do método Galerkin Descontínuo e na aplicação da aproximação Runge-Kutta de terceira ordem no tempo. A formulação do método Galerkin Descontínuo utiliza uma abordagem local, visto que a abordagem global não é capaz de lidar com a formação de choque e a interação das ondas. A utilização deste método requer a aplicação do limitador TVB para retirar as oscilações espúrias geradas, e no caso em que a variável conservada/balanceada é limitada será necessário também a aplicação do limitador positive preserving. Para calcular as integrais requeridas uma quadratura Gaussiana especial foi aplicada, de forma a satisfazer a condição CFL que garante a convergência do método numérico.

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Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian–Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems

Abreu

Santo

Lambert

et al. 2022

Numerical Methods Partial

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A Class of Lagrangian–Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms

Cited by 17 publications

References 55 publications

Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

Aplicação de esquemas Galerkin Descontínuo em problemas hiperbólicos em 2D

Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian–Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems

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