Fully adaptive multiresolution finite volume schemes for conservation laws

Abstract

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.

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

“…An error analysis, which has been adapted from Cohen et al [11] and is also advanced in [59] for strongly degenerate parabolic equations of the type (1.4), is presented in Section 5. This error analysis motivates the choice of two parameters in the thresholding algorithm.…”

“…Following the ideas introduced by Cohen et al [11] and thereafter extended to parabolic equations by Roussel et al [9] we decompose the global error between the cell average values of the exact solution at the level L, denoted byū L ex , and those of the multiresolution computation with a maximal level L, denoted byū…”

“…For the second error, called perturbation error, Cohen et al [11] assume that the details on a level l are deleted when smaller than a prescribed tolerance ε l . Under this assumption, they show that if the numerical scheme, i.e.…”

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

“…Hence, the main idea is to use the decay of the wavelet coefficients to obtain information on local regularity of the solution: lower wavelet coefficients are associated to local regular spatial configurations and vice-versa. This representation yields to a thresholding process that builds dynamically the corresponding adapted grid on which the solutions are represented; then the error committed by the multiresolution transformation is proportional to η MR , where η MR is a threshold parameter [9,2].…”

“…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.…”