Adaptive multiresolution computations applied to detonations

A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and dynamic space adaptivity is introduced using multiresolution analysis. A time splitting method of Strang is applied to be able to consider stiff problems while keeping the method explicit. For time adaptivity an improved Runge--Kutta--Fehlberg scheme is used. Applications deal with detonation problems in one and two space dimensions. A comparison of the adaptive scheme with reference computations on a regular grid allow to assess the accuracy and the computational efficiency, in terms of CPU time and memory requirements.Comment: Zeitschrift f\"ur Physicalische Chemie, accepte

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.Comment: 27 pages, 14 figure

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

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 Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the 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

Wavelets And Adaptive Grids For The Discontinuous Galerkin Method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method. © Springer 2005.391-3143154Abgrall, R., Harten, A., Multiresolution representation in unstructured meshes (1998) SIAM J. Numer. Anal.Bihari, B.L., Harten, A., Multiresolution schemes for the numerical solution of 2-D conservation laws I (1997) SIAM J. Sci. Comput., 18 (2)Bonhaus, D.L., (1998) A Higher Order Accurate Finite Element Method for Viscous Compressible Flows, , Ph.D. thesis, Virginia Polytechnics Institute and State University (November)Brooks, A., Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations (1982) Comput. Methods Appl. Mech. Engrg., 32Chiavassa, G., Donat, R., Numerical experiments with multilevel schemes for conservation laws (1999) Godunov's Methods: Theory and Applications, , ed. Toro (Kluwer Academic/Plenum, Dordrecht)Cockburn, B., Shu, C.-W., Runge-Kutta discontinuous Galerkin method for convection-dominated problems (2001) J. Sci. Comput., 16Cohen, A., Muller, S., Postel, M., Ould-Kabe, S.M., Fully adaptive multiresolution finite volume schemes for conservation laws (2002) Math. Comp., 72Dahmen, W., Gottschlich-Müller, B., Müller, S., Multiresolution schemes for conservation laws (1998) Numer. Math., 88Díaz Calle, J.L., Devloo, P.R.B., Gomes, S.M., Stabilized discontinuous Galerkin method for hyperbolic equations Comput. Methods Appl. Mech. Engrg., , to appearDomingues, M.O., Gomes, S.M., Diaz, L.A., Adaptive wavelet representation and differentiation on block-structured grids (2003) Appl. Numer. Math., 8 (3-4)Harten, A., Adaptive multiresolution schemes for shock computations (1994) J. Comput. Phys., 115Harten, A., Multiresolution representation of data: A general framework (1996) SIAM J. Numer. Anal., 33Holmström, M., (1997) Wavelet Based Methods for Time Dependent PDE, , Ph.D. thesis, Uppsala University, SwedenKaibara, M.K., Gomes, S.M., Fully adaptive multiresolution scheme for shock computations (1999) Godunov's Methods: Theory and Applications, , ed. Toro (Kluwer Academic/Plenum, Dordrecht)Sjögreen, B., Numerical experiments with the multiresolution schemes for the compressible Euler equations (1995) J. Comput. Phys., 117Vasilyev, O.V., Bowman, C., Second generation wavelet collocation method for the solution of partial differential equations (2000) J. Comput. Phys., 165Waldén, J., Filter bank methods for hyperbolic PDEs (1999) SIAM J. Numer. Anal., 3