Multiresolution Schemes on Triangles for Scalar Conservation Laws

Cohen, Albert; Dyn, Nira; Kaber, Sidi Mahmoud; Postel, Marie

doi:10.1006/jcph.2000.6503

Cited by 40 publications

(21 citation statements)

References 14 publications

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“…In this context, particular adaptations of the prediction operator (based on stable completion techniques introduced in [6]) are needed to preserve accuracy. In the case of uniform triangular discretization, finite volume multiresolution with accuracy, stability, and smoothness properties has been obtained in [13]. In the case of unstructured meshes much less is known: while finite volume multiresolution can be designed with accuracy properties (see [1]), a general strategy to achieve stability and smoothness properties is still an open question.…”

Section: Discussionmentioning

confidence: 99%

“…This information is used to accelerate the scheme by saving on the evaluation of the numerical flux, which is exactly computed on the finest mesh only in the regions of poor smoothness (or high gradients), otherwise computed approximately from its exact computation on coarser meshes. While this strategy was initially developed for one-dimensional structured grids, several contributions have made it operational for various types of multivariate finite volume meshes (Cartesian [5,8], curvilinear [16,30], triangular [13,31] and unstructured [1]). Several remarks should be made concerning such a strategy:…”

Section: Multiresolution Methodsmentioning

confidence: 99%

“…Here we use the prediction operator introduced in [13] and the direct evaluation strategy for the adaptive algorithm. Some first numerical tests for a linear advection problem have been presented in [14] together with a detailed description of the algorithm.…”

Section: Tests In 2dmentioning

confidence: 99%

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Fully adaptive multiresolution finite volume schemes for conservation laws

et al. 2001

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Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

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Section: Discussionmentioning

confidence: 99%

Section: Multiresolution Methodsmentioning

confidence: 99%

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Fully adaptive multiresolution finite volume schemes for conservation laws

et al. 2001

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“…This so-called fully adaptive scheme has been analysed in the explicit scalar case on Cartesian grids in [6] and further implemented in the system cases, for instance, in [7,18,19].…”

Section: Adaptive Multiresolutionmentioning

confidence: 99%

A relaxation multiresolution scheme for accelerating realistic two‐phase flows calculations in pipelines

Andrianov

Coquel

Postel

et al. 2006

Numerical Methods in Fluids

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SUMMARYWe wish to demonstrate that it is judicious to combine various existing computational techniques that appeared for academic cases in seemingly unrelated areas, namely, semi-implicit relaxation schemes for hyperbolic systems and adaptive multiresolution algorithms, in order to achieve fast and accurate simulations of realistic two-phase flows problems in oil transportation. By 'realistic' we mean problems that are modelled by partial differential equation (PDE) systems closed by sophisticated thermodynamics and hydrodynamics laws, set out over a terrain-induced geometry and associated with time-dependent boundary conditions. Although the combination of these techniques is not a straightforward matter, it is made possible via a careful examination of the objectives of the simulation problem and suitable adaptations of which we shall give the details. Significant benchmarks demonstrate the efficiency of the proposed method.

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“…Numerical methods based on 879 et al [12], by constructing a multiresolution scheme using triangles, have also shown that it is not necessary to perform the computation on a cartesian grid. Bramkamp et al [13] used parametric meshes to map grid cells to splines enabling them to discretize the fluid surrounding an aerofoil.…”

Section: Introductionmentioning

confidence: 98%

Wavelet‐based adaptive grids as applied to hydrodynamics

Smith¹,

Zang²,

Taylor³

2007

Numerical Methods in Fluids

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SUMMARYAn adaptive, wavelet-based, multiscale finite-volume scheme is developed and employed to investigate applications in the simulation of water waves. Firstly, two one-dimensional, strictly hyperbolic cases are investigated: shallow water and Euler equations. These are followed by two investigations using a finitevolume formulation of Madsen and Sørensen's Boussinesq equations. Converged results were obtained in all cases, which demonstrate that the adaptive grid scheme is significantly faster than that on a uniform grid. In some cases, one-seventh of the number of cells is required to obtain the same accuracy as that of the uniform grid. Issues of stability are discussed in the context of the particular problems modelled here with the Boussinesq equations, related to discretization of the high-order spatial derivatives on a non-uniform grid.

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Multiresolution Schemes on Triangles for Scalar Conservation Laws

Cited by 40 publications

References 14 publications

Fully adaptive multiresolution finite volume schemes for conservation laws

Fully adaptive multiresolution finite volume schemes for conservation laws

A relaxation multiresolution scheme for accelerating realistic two‐phase flows calculations in pipelines

Wavelet‐based adaptive grids as applied to hydrodynamics

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