Adaptive Multiscale Schemes for Conservation Laws

Müller, Siegfried

doi:10.1007/978-3-642-18164-1

Cited by 136 publications

(218 citation statements)

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“…It goes back to Harten [7] for hyperbolic equations and was used by Bihari and Harten [8] and Roussel et al [9] for parabolic equations. Important contributions to the analysis of multiresolution methods for conservation laws include [10,11,12].…”

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confidence: 99%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Bürger

Ruiz

Schneider³

et al. 2007

J Eng Math

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Abstract. A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the approximation of Engquist-Osher for the flux and explicit time stepping. An adaptive multiresolution scheme with cell averages is then used to speed up CPU time and memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier-thickener model illustrate the efficiency of this method.

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confidence: 99%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Bürger

Ruiz

Schneider³

et al. 2007

J Eng Math

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“…We refer to Müller [23] for a survey on MR methods, see also Chiavassa et al [24], and Cohen et al [8] and Dahmen et al [25] for the application of classical MR methods to hyperbolic partial differential equations.…”

Section: B Related Workmentioning

confidence: 99%

“…In the context of fully adaptive MR methods [8], the mathematical analysis is complete only in the case of a scalar conservation law, but in practice, these techniques have been used by several groups (see e.g., [7,9,10,23,26]) to successfully solve a wide class of problems, including applications to multidimensional systems. These results illustrate that the MR method has turned out to be a useful device for a series of problems with a similar structure to cardiac problems.…”

Section: B Related Workmentioning

confidence: 99%

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A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

Bendahmane¹,

Bürger²,

Ruiz–Baier³

2009

Numerical Methods Partial

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The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.

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“…Moreover, this splitting time step is dynamically adapted taking into account local error estimates [4]. The time integration is performed over a dynamic adapted grid obtained by multiresolution techniques in a finite volumes framework [9,2,11], which on the one hand, yield important savings in computing resources and on the other hand, allow to somehow control the spatial accuracy of the compressed representation based on a solid mathematical background.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Time-Space Algorithms for the Simulation of Multi-scale Reaction Waves

Duarte

Massot

Descombes³

et al. 2011

Finite Volumes for Complex Applications VI Problems &Amp; Perspectives

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We present a new resolution strategy for multi-scale reaction waves based on adaptive time operator splitting and space adaptive multiresolution, in the context of localized and stiff reaction fronts. The main goal is to perform computationally efficient simulations of the dynamics of multi-scale phenomena under study, considering large simulation domains with conventional computing resources. We aim at time-space accuracy control of the solution and splitting time steps purely dictated by the physics of the phenomenon and not by stability constraints associated with mesh size or source time scales. Numerical illustrations are provided for 2D and 3D combustion applications modeled by reaction-convection-diffusion equations.

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Adaptive Multiscale Schemes for Conservation Laws

Cited by 136 publications

References 0 publications

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

Adaptive Time-Space Algorithms for the Simulation of Multi-scale Reaction Waves

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