A new finite volume approach for transport models and related applications with balancing source terms

Abreu, Eduardo; Lambert, Wanderson; Pérez, J. M.; Santo, Arthur Espírito

doi:10.1016/j.matcom.2016.12.012

Cited by 28 publications

(46 citation statements)

References 30 publications

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“…We consider a novel Lagrangian-Eulerian formulation that can be viewed as an extension of previous works [2,3,8] for practical construction of numerical solutions for balance law problems, but following innovative recent ideas introduced in [1] to construct weak asymptotic methods for scalar equations and systems of conservation law equations. For simplicity, we consider the particular scalar equation with u = u(x, t)…”

Section: Methodsmentioning

confidence: 99%

“…3 As in [3], the curves σ n j−1/2 (t) are not straight lines in general, but rather solutions of the local nonlinear differential equations [2,4,8]:…”

Section: Methodsmentioning

confidence: 99%

“…The extension of this construction follows naturally from the finite volume formulation of the linear Lagrangian-Eulerian scheme, as in [2,3,8], building block to construct local approximations such as…”

Section: Methodsmentioning

confidence: 99%

“…The Lagrangian-Eulerian approach is a promising tool for numerically solving partial differential equations of several types. Recently, in [2,3,8] such ideas were extended to a wide range of nonlinear purely hyperbolic conservation laws and balance laws scalar and systems. Here, we make use of polynomial reconstruction ideas into the Lagrangian-Eulerian novel approach, but keeping the scheme simple, fast and without any strong restriction over the source term other than integrability on the finite volume.…”

Section: Introductionmentioning

confidence: 99%

“…Here, we make use of polynomial reconstruction ideas into the Lagrangian-Eulerian novel approach, but keeping the scheme simple, fast and without any strong restriction over the source term other than integrability on the finite volume. As in [2,3,8] the hyperbolic part of balance law is written in a space time divergence form so that the inherent conservation properties of the hyperbolic operator are used efficiently to build a numerical method to hyperbolic balance law problems [5]. Such framework presents an interesting property of being rather independent of a particular structure of the source terms.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A conservative Lagrangian-Eulerian finite volume approximation method for balance law problems

Abreu¹,

Pérez²,

Santo³

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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We present a simple numerical method based on a Lagrangian-Eulerian framework for approximate solutions of nonlinear balance law problems. This framework has been used for numerically solving partial differential equations of several types, such as hyperbolic conservation laws [3,8], balance laws problems [4]. As in [3,5] the mass conservation takes place in the space-time volume D n j , and this region in the form of [3] is used to define naturally a balance law. This balance law is the central idea to build a efficient numerical method to approximate solution to balance law problems. Verification of the technique is also made by comparison with analytical solutions when they are available.

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Section: Methodsmentioning

confidence: 99%

“…3 As in [3], the curves σ n j−1/2 (t) are not straight lines in general, but rather solutions of the local nonlinear differential equations [2,4,8]:…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A conservative Lagrangian-Eulerian finite volume approximation method for balance law problems

Abreu¹,

Pérez²,

Santo³

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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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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An Enhanced Lagrangian‐Eulerian Method for a Class of Balance Laws: Numerical Analysis via a Weak Asymptotic Method With Applications

Abreu,

Pandini,

Lambert

2024

Numerical Methods Partial

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In this work, we designed and implemented an enhanced Lagrangian‐Eulerian numerical method for solving a wide range of nonlinear balance laws, including systems of hyperbolic equations with source terms. We developed both fully discrete and semi‐discrete formulations, and extended the concept of No‐Flow curves to this general class of nonlinear balance laws. We conducted a numerical convergence study using weak asymptotic analysis, which involved investigating the existence, uniqueness, and regularity of entropy‐weak solutions computed with our scheme. The proposed method is Riemann‐solver‐free. To evaluate the shock‐capturing capabilities of the enhanced Lagrangian‐Eulerian numerical scheme, we carried out numerical experiments that demonstrate its ability to accurately resolve the key features of balance law models and hyperbolic problems. A representative set of numerical examples is provided to illustrate the accuracy and robustness of the proposed method.

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A new finite volume approach for transport models and related applications with balancing source terms

Cited by 28 publications

References 30 publications

A conservative Lagrangian-Eulerian finite volume approximation method for balance law problems

A conservative Lagrangian-Eulerian finite volume approximation method for balance law problems

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

An Enhanced Lagrangian‐Eulerian Method for a Class of Balance Laws: Numerical Analysis via a Weak Asymptotic Method With Applications

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