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

(Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models: robust 2D approaches

Multiwavelets (MW) enable the compression, analysis and assembly of model data on a multiresolution grid within Godunov-type solvers based on second-order discontinuous Galerkin (DG2) and first-order finite volume (FV1) methods. Multiwavelet adaptivity has been studied extensively with one-dimensional (1D) hydrodynamic models (Kesserwani et al., 2019), revealing that MWDG2 can be 20 times faster than uniform DG2 and 2 times faster than uniform FV1, while preserving the accuracy and robustness of the underlying formulation. The potential of the MWDG2 scheme has yet to be studied for two-dimensional (2D) modelling, but this requires a design that robustly and efficiently solves the 2D shallow water equations (SWE) with complex source terms and wetting and drying. This paper presents a two-dimensional MWDG2 scheme that: (1) adopts a slope-decoupled DG2 solver as a reference scheme, for its ability to deliver well-balanced piecewise-planar solutions shaped by a simplified 3-component basis; and, (2) adapts the multiresolution analysis of multiwavelets for compatibility with the slope-decoupled DG2 basis. A scaled reformulation of slope-decoupled DG2 is presented alongside two multiwavelet approaches that yield MWDG2 schemes with similar properties, and a Haar wavelet FV1 (HFV1) variant for adapting piecewise-constant model data. The performance of the adaptive HFV1 and MWDG2 solvers is explored alongside their uniform counterparts, while analysing their accuracy, efficiency, grid-coarsening ability, reliability in handling wet-dry fronts across steep bed-slopes, and ability to capture features relevant to practical hydraulic modelling. The results indicate a particular multiwavelet approach that allows the MWDG2 scheme to exploit its grid-coarsening ability for the widest range of flow types. Results also indicate that the proposed (multi)wavelet-based adaptive schemes are even more efficient for the 2D case. Accompanying model software is openly available online

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