A high-order finite volume method for systems of conservation laws—Multi-dimensional Optimal Order Detection (MOOD)

Clain, Stéphane; Diot, Steven; Loubère, Raphaël

doi:10.1016/j.jcp.2011.02.026

Cited by 277 publications

(351 citation statements)

References 26 publications

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“…Very recently, a promising alternative has been proposed by Dumbser et al (2014), which is based on a previous idea of Clain et al (2011) and Diot et al (2012) called MOOD (multi-dimensional optimal order detection), and which adopts an a posteriori approach to the problem of limiting of high order schemes in the finite volume framework. In a few words, the novel a posteriori DG limiter method of Dumbser et al (2014) consists of (i) computing the solution by means of an unlimited ADER-DG scheme, (ii) detecting a posteriori the troubled cells by applying a simple discrete maximum principle (DMP) and positivity of density and pressure on the discrete solution, (iii) creating a local sub-grid within these troubled DG cells, and (iv) recomputing the discrete solution at the sub-grid level via a more robust Total Variation Diminishing (TVD) or Weighted Essentially Non Oscillatory (WENO) finite volume scheme.…”

Section: Introductionmentioning

confidence: 99%

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

Zanotti

Fambri

Dumbser

2015

Mon. Not. R. Astron. Soc.

114

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We present a new numerical tool for solving the special relativistic ideal MHD equations that is based on the combination of the following three key features: (i) a one-step ADER discontinuous Galerkin (DG) scheme that allows for an arbitrary order of accuracy in both space and time, (ii) an a posteriori subcell finite volume limiter that is activated to avoid spurious oscillations at discontinuities without destroying the natural subcell resolution capabilities of the DG finite element framework and finally (iii) a space-time adaptive mesh refinement (AMR) framework with time-accurate local time-stepping.The divergence-free character of the magnetic field is instead taken into account through the so-called "divergence-cleaning" approach. The convergence of the new scheme is verified up to 5 th order in space and time and the results for a set of significant numerical tests including shock tube problems, the RMHD rotor and blast wave problems, as well as the Orszag-Tang vortex system are shown. We also consider a simple case of the relativistic Kelvin-Helmholtz instability with a magnetic field, emphasizing the potential of the new method for studying turbulent RMHD flows. We discuss the advantages of our new approach when the equations of relativistic MHD need to be solved with high accuracy within various astrophysical systems.

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Section: Introductionmentioning

confidence: 99%

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

Zanotti

Fambri

Dumbser

2015

Mon. Not. R. Astron. Soc.

114

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“…We dynamical determine the CPD map using a a posteriori MOOD-like approach developed in [13,16,17]. Namely, for a given stage k and its associated map M k , a candidate solution Φ k is computed.…”

Section: Mood Loopmentioning

confidence: 99%

“…For example, the positivity preserving is a consequence of the TVD restriction on the reconstruction but is not directly imposed as a restriction per se. The Multidimensional Optimal Order Method (MOOD) has been designed in [13,16,17] on different paradigms. It is an a posteriori method since the modification of the polynomial reconstruction is performed after calculating a candidate solution.…”

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confidence: 99%

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

2017

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To cite this version:Stéphane Clain, Raphaël Loubère, Gaspar Machado. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. 2016. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equationsWe propose a new family of finite volume high-accurate numerical schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers', and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

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“…The technique is based on specific polynomial reconstructions used for the fluxes (Hernández, 2002;Ollivier-Gooch and Altena, 2002;Toro and Hidalgo, 2009;Toro, 2009;Clain et al, 2011; The organization of the paper is the following. Section 2 is devoted to the harmonic operator, where we introduce the mesh, the generic finite volume formulation, and the polynomial reconstruction operator to design the high-order finite volume scheme.…”

Section: Introductionmentioning

confidence: 99%

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

Costa

Machado

Clain

2015

International Journal of Applied Mathematics and Computer Science

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A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.

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A high-order finite volume method for systems of conservation laws—Multi-dimensional Optimal Order Detection (MOOD)

Cited by 277 publications

References 26 publications

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

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