ADER schemes on adaptive triangular meshes for scalar conservation laws

Käser, Martin; Iske, Armin

doi:10.1016/j.jcp.2004.11.015

Cited by 119 publications

(133 citation statements)

References 27 publications

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“…The required exchange of information between neighbouring particles is modelled via a generic numerical flux function, which may be implemented by using any suitable FV flux evaluation scheme, such as ADER in [8]. For details on further features of FVPM, see [3,7].…”

Section: Finite Volume Particle Methodsmentioning

confidence: 99%

“…It is sufficient for the purpose of this short contribution to say that a particle ξ ∈ Ξ is coarsened by its removal from Ξ, whereas ξ is refined by the insertion of further particles in its neighbourhood. For further details concerning computational aspects of the utilized adaption rules and their construction, we refer to our previous papers [5,8].…”

Section: Adaption Rulesmentioning

confidence: 99%

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Multiscale Flow Simulation by Adaptive Finite Volume Particle Methods

Iske

2005

Proc Appl Math and Mech

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We report on recent developments concerning adaptive finite particle methods, which are used for the numerical simulation of multiscale phenomena in time-dependent evolution processes. The proposed concept relies on a finite volume approach, which we combine with WENO reconstruction from particle average values. In this method, polyharmonic splines are key tools for both the optimal recovery from scattered particle averages and the construction of customized adaption rules.

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Section: Finite Volume Particle Methodsmentioning

confidence: 99%

Section: Adaption Rulesmentioning

confidence: 99%

Multiscale Flow Simulation by Adaptive Finite Volume Particle Methods

Iske

2005

Proc Appl Math and Mech

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“…Let us further assume that j i (1) = i. In order to reach the nominal order of accuracy M + 1, we must choose n e (i) ≥ (M + 1)(M + 2)/2, see [BF90,OGA02,KI05]. In our case we include in S i all the cells intersecting Ω i at least in a point (see [SCR15] for a proof that there are at least 5 neighbors in a generic quad-tree mesh and for a discussion of the three-dimensional generalization).…”

Section: Polynomial Reconstruction Operatormentioning

confidence: 99%

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

Semplice

Loubère

2018

Journal of Computational Physics

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To cite this version:Matteo Semplice, Raphaël Loubère. Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions. AbstractIn this paper we propose a third order accurate finite volume scheme based on polynomial reconstruction along with a posteriori limiting within an Adaptive-Mesh-Refinement (AMR) simulation code for hydrodynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge-Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a nonoscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using this time a more dissipating scheme but only locally to these cells. By decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate scheme. Ultimately some troubled cells may be updated with a first order accurate scheme for instance close to steep gradients. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed but only locally with such refined mesh. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology not only can maintain optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context.

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“…Further generalisations were put forward by Dumbser et al [12], setting ADER-FV and ADER-DG in a generalised framework. The ADER approach has undergone numerous extensions and applications, examples include [2,21,23,22,35,36,37,40,41,12,5,6,10,11,3,9,7,1,28,29,31,13,14,30]. A succinct review of the ADER approach, in the frame of the finite volume method, is presented here, in terms of a one-dimensional system of hyperbolic balance laws…”

Section: The Ader Approachmentioning

confidence: 99%

A novel numerical flux for the 3D Euler equations with general equation of state

Toro

Castro

Lee

2015

Journal of Computational Physics

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Please cite this article in press as: E. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractHere we extend the flux vector splitting approach recently proposed in (E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012). The scheme was originally presented for the 1D Euler equations for ideal gases and its extension presented in this paper is threefold: (i) we solve the three-dimensional Euler equations on general meshes; (ii) we use a general equation of state; and (iii) we achieve high order of accuracy in both space and time through application of the semi-discrete ADER methodology on general meshes. The resulting methods are systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems. Formal high accuracy is assessed through convergence rates studies for schemes of up to 4th order of accuracy in both space and time on unstructured meshes.

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ADER schemes on adaptive triangular meshes for scalar conservation laws

Cited by 119 publications

References 27 publications

Multiscale Flow Simulation by Adaptive Finite Volume Particle Methods

Multiscale Flow Simulation by Adaptive Finite Volume Particle Methods

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

A novel numerical flux for the 3D Euler equations with general equation of state

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