Space-time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations. (English) Zbl 1165.76031
Summary: Adaptive strategies in space and time allow considerable speed-up of finite volume schemes for conservation laws, while controlling the accuracy of the discretization. In this paper, we present a multiresolution technique for finite volume schemes with explicit time discretization. An adaptive grid is introduced by suitable thresholding of wavelet coefficients, which maintains the accuracy of the finite volume scheme on a regular grid. Further speed-up is obtained by local scale-dependent time stepping, i.e., on large scales larger time steps can be used without violating the stability condition of the explicit scheme. Furthermore, an estimation of the truncation error in time, using embedded Runge-Kutta type schemes, guarantees a control of the time step for a given precision. The accuracy and efficiency of the fully adaptive methods is illustrated by applications to compressible Euler equations in one and two space dimensions.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76N15 | Gas dynamics (general theory) |
Software:
AUSMReferences:
| [1] | Abgrall, R., Multiresolution analysis on unstructured meshes: applications to CFD, (Chetverushkin, B. E.A., Experimentation, Modelling and Computation in Flow, Turbulence and Combustion (1997), Wiley) · Zbl 0925.76421 |
| [2] | Alam, J.; Kevlahan, N.-K.-R.; Vasilyev, O., Simultaneous space-time adaptive wavelet solution of nonlinear partial differential equations, J. Comput. Phys., 214, 829-857 (2006) · Zbl 1089.65103 |
| [3] | Bacry, E.; Mallat, S.; Papanicolaou, G., A wavelet based space-time adaptive numerical-method for partial-differential equations, RAIRO Math. Modell. Numer. Anal., 26, 793-834 (1992) · Zbl 0768.65062 |
| [4] | Berger, M. J.; Collela, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 67-84 (1989) · Zbl 0665.76070 |
| [5] | Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484-512 (1984) · Zbl 0536.65071 |
| [6] | Bihari, B. L., Multiresolution schemes for conservation laws with viscosity, J. Comput. Phys., 123, 207-225 (1996) · Zbl 0840.65093 |
| [7] | Bihari, B. L.; Harten, A., Multiresolution schemes for the numerical solution of 2-D conservation laws I, SIAM J. Sci. Comput., 18, 2, 315-354 (1997) · Zbl 0878.35007 |
| [8] | Calle, J. D.; Devloo, P. R.B.; Gomes, S. M., Wavelets and adaptive grids for the discontinuous Galerkin method, Numer. Algorithms, 39, 143-158 (2005) · Zbl 1068.65118 |
| [9] | Chiavassa, G.; Donat, R., Point value multi-scale algorithms for 2D compressible flow, SIAM J. Sci. Comput., 23, 3, 805-823 (2001) · Zbl 1043.76046 |
| [10] | Cohen, A., Wavelet methods in numerical analysis, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis, vol. VII (2000), Elsevier: Elsevier Amsterdam) · Zbl 0976.65124 |
| [11] | Cohen, A.; Kaber, S. M.; Müller, S.; Postel, M., Fully adaptive multiresolution finite volume schemes for conservation laws, Math. Comp., 72, 183-225 (2003) · Zbl 1010.65035 |
| [12] | Collino, F.; Fouquet, T.; Joly, P., A conservative space-time mesh refinement method for the 1-D wave equation. I. Construction, Numer. Math., 95, 2, 197-221 (2003) · Zbl 1048.65089 |
| [13] | Dawson, C.; Kirby, R., High resolution schemes for conservation laws with locally varying time steps, SIAM J. Sci. Comput., 22, 6, 2256-2281 (2001) · Zbl 0980.35015 |
| [14] | Domingues, M. O.; Gomes, S. M.; Diaz, L. M.A., Adaptive wavelet representation and differentiation on block-structured grids, Appl. Numer. Math., 47, 421-437 (2003) · Zbl 1035.65167 |
| [15] | Domingues, M. O.; Gomes, S. M.; Roussel, O.; Schneider, K., An adaptive multiresolution scheme with local time-stepping for evolutionary PDEs, J. Comput. Phys., 227, 3758-3780 (2008) · Zbl 1139.65060 |
| [16] | Domingues, M. O.; Roussel, O.; Schneider, K., On space-time schemes for the numerical solution of PDEs, ESAIM Proc., 16, 181-194 (2007) · Zbl 1206.65228 |
| [17] | M. O. Domingues, O. Roussel, K. Schneider, An adaptive multiresolution method for parabolic PDEs with time control, Int. J. Numer. Meth. Eng., DOI: 10.1002/nme.2501; M. O. Domingues, O. Roussel, K. Schneider, An adaptive multiresolution method for parabolic PDEs with time control, Int. J. Numer. Meth. Eng., DOI: 10.1002/nme.2501 · Zbl 1183.76816 |
| [18] | Dumbser, M.; Käser, M.; Toro, E., An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes V: Local time stepping and p-adaptivity, Geophys. J. Int., 171, 2, 695-717 (2006), (23) |
| [19] | Ferm, L.; Löstedt, P., Space-time adaptive solutions of first order PDEs, J. Sci. Comput., 26, 1, 83-110 (2006) · Zbl 1089.76041 |
| [20] | Flaherty, J. E.; Loy, R. M.; Shephard, M. S.; Szymanski, B. K.; Teresco, J. D.; Ziantz, L. H., Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws, J. Parall. Dist. Comp., 47, 2, 139-152 (1997) |
| [21] | Gottschlich-Müller, B.; Müller, S., Adaptive finite volume schemes for conservation laws based on local multiresolution techniques, (Jeltsch, R.; Fey, M., Hyperbolic Problems: Theory, Numerics, Applications. Hyperbolic Problems: Theory, Numerics, Applications, Inter. Ser. Numer. Math., ISNM, vol. 129 (1999), Birkhäuser Verlag Bassel: Birkhäuser Verlag Bassel Switzerland, Bassel) · Zbl 0929.65081 |
| [22] | Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, Computational Mathematics, vol. 8 (2000), Springer: Springer Berlin |
| [23] | Harten, A., Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Comm. Pure Appl. Math., 48, 1305-1342 (1995) · Zbl 0860.65078 |
| [24] | Harten, A., Multiresolution representation of data: a general framework, SIAM, J. Numer. Anal., 33, 3, 385-394 (1996) |
| [25] | M. Holmström, Wavelet based methods for time dependent PDEs, Ph.D. thesis, Uppsala University, 1997; M. Holmström, Wavelet based methods for time dependent PDEs, Ph.D. thesis, Uppsala University, 1997 |
| [26] | Holmström, M., Solving hyperbolic PDEs using interpolating wavelets, SIAM J. Sci. Comput., 21, 2, 405-420 (1999) · Zbl 0959.65109 |
| [27] | Hörnel, K.; Lötstedt, P., Time step selection for shock problems, Comm. Numer. Meth. Eng., 17, 477-484 (2001) · Zbl 0985.65115 |
| [28] | Kaibara, M.; Gomes, S. M., A fully adaptive multiresolution scheme for shock computations, (Toro, E. F., Godunov Methods: Theory and Applications (2000), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers New York) · Zbl 1064.76589 |
| [29] | Keppens, R.; Nool, M.; Tóth, G.; Goedbloed, T. P., Adaptive mesh refinement for conservative systems: multi-dimensional efficiency evaluation, Comput. Phys. Comm., 153, 317-339 (2003) · Zbl 1196.76055 |
| [30] | Lamby, P.; Massjung, R.; Müller, S.; Stiriba, Y., Inviscid flow on moving grids with multiscale space and time adaptivity, (Bermudez de Castro, A.; Gomez, D.; Quintela, P.; Salgado, P., Numerical Mathematics and Advanced Applications (2006), Springer: Springer Berlin, Heidelberg), 831-839 · Zbl 1278.76069 |
| [31] | Lamby, P.; Müller, S.; Stiriba, Y., Solution of shallow water equations using fully adaptive multiscale schemes, Int. J. Numer. Meth. Fluids, 49, 4, 417-437 (2005) · Zbl 1086.76050 |
| [32] | Leveque, R. J., Finite Volume Methods for Hyperbolic Systems (2002), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1010.65040 |
| [33] | Liou, M.-S., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 364-382 (1996) · Zbl 0870.76049 |
| [34] | F. Lörcher, G. Gassner, C.-D. Munz, A discontinuous Galerkin scheme based on a space-time expansion I. inviscid compressible flow in one space dimension, J. Sci. Comput. 32 (2), 175-199; F. Lörcher, G. Gassner, C.-D. Munz, A discontinuous Galerkin scheme based on a space-time expansion I. inviscid compressible flow in one space dimension, J. Sci. Comput. 32 (2), 175-199 · Zbl 1143.76047 |
| [35] | Müller, S., Adaptive Multiscale Schemes for Conservation Laws, Lectures Notes in Computational Science and Engineering, vol. 27 (2003), Springer: Springer Heidelberg · Zbl 1016.76004 |
| [36] | Müller, S.; Stiriba, Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping, J. Sci. Comput., 30, 3, 493-531 (2007) · Zbl 1110.76037 |
| [37] | Osher, S.; Sanders, R., Numerical approximations to nonlinear conservation laws with locally varying time space grid, Math. Comp., 43, 321-336 (1983) · Zbl 0592.65068 |
| [38] | Pinho, P.; Domingues, M. O.; Ferreira, P. J.; Gomes, S. M.; Gomide, A.; Pereira, J. R., Interpolating wavelets and adaptive finite difference schemes for solving Maxwell’s equations: the effects of gridding, IEEE Trans. Magnetics, 43, 1013-1022 (2007) |
| [39] | Roussel, O.; Schneider, K., An adaptive multiresolution method for combustion problems: application to flame ball – vortex interaction, Comput. Fluids, 34, 7, 817-831 (2005) · Zbl 1134.80304 |
| [40] | Roussel, O.; Schneider, K.; Tsigulin, A.; Bockhorn, H., A conservative fully adaptive multiresolution algorithm for parabolic PDEs, J. Comput. Phys., 188, 493-523 (2003) · Zbl 1022.65093 |
| [41] | Shampine, L. F., Error estimation and control of ODEs, J. Sci. Comput., 25, 112, 3-15 (2005) · Zbl 1203.65122 |
| [42] | Shampine, L. F.; Witt, A., Control of local error stabilizes integrations, J. Comput. Appl. Math., 62, 333-351 (1995) · Zbl 0858.65087 |
| [43] | Sommeijer, B. P.; Shampine, L. F.; Vewer, G. J., An explicit solver for parabolic PDEs, J. Comput. Appl. Math., 88, 3, 315-326 (1997) · Zbl 0910.65067 |
| [44] | Tang, H. Z.; Warnecke, G., A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and space grids, SIAM J. Sci. Comput., 26, 4, 1415-1431 (2005) · Zbl 1079.65092 |
| [45] | Wesseling, P., Principles of Computational Fluid Dynamics (2001), Springer: Springer Berlin · Zbl 0989.76069 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
