article

High-order explicit local time-stepping methods for damped wave equations

Authors:

Marcus J. Grote,

Teodora MitkovaAuthors Info & Claims

Journal of Computational and Applied Mathematics, Volume 239

Pages 270 - 289

https://doi.org/10.1016/j.cam.2012.09.046

Published: 01 February 2013 Publication History

Abstract

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

References

[1]

Ciarlet, P., The Finite Element Method for Elliptic Problems. 1978. North-Holland, Amsterdam.

[2]

Hughes, T., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. 2000. Dover Publications.

[3]

Cohen, G., Joly, P., Roberts, J. and Tordjman, N., Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. v38. 2047-2078.

Digital Library

[4]

Giraldo, F.X. and Taylor, M.A., A diagonal-mass-matrix triangular-spectral-element method based on cubature points. J. Engrg. Math. v56. 307-322.

[5]

TVB Runge-Kutta local projection discontinuous Galerkin method for conservation laws II: general framework. Math. Comp. v52. 411-435.

[6]

Discontinuous finite element methods for acoustic and elastic wave problems. Contemp. Math. v329. 271-282.

[7]

Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. v27. 5-40.

Digital Library

[8]

Grote, M.J., Schneebeli, A. and Schötzau, D., Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. v44. 2408-2431.

[9]

Grote, M.J., Schneebeli, A. and Schötzau, D., Interior penalty discontinuous Galerkin method for Maxwell's equations: energy norm error estimates. J. Comput. Appl. Math. v204. 375-386.

Digital Library

[10]

Nodal Discontinuous Galerkin Methods. 2008. Springer.

[11]

Collino, F., Fouquet, T. and Joly, P., A conservative space-time mesh refinement method for the 1-D wave equation. I. Construction. Numer. Math. v95. 197-221.

[12]

Collino, F., Fouquet, T. and Joly, P., Conservative space-time mesh refinement method for the FDTD solution of Maxwell's equations. J. Comput. Phys. v211. 9-35.

Digital Library

[13]

Dolean, V., Fahs, H., Fezoui, L. and Lanteri, S., Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys. v229. 512-526.

Digital Library

[14]

Piperno, S., Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems. M2AN Math. Model. Numer. Anal. v40. 815-841.

[15]

Montseny, E., Pernet, S., Ferriéres, X. and Cohen, G., Dissipative terms and local time-stepping improvements in a spatial high order discontinuous Galerkin scheme for the time-domain Maxwell's equations. J. Comput. Phys. v227. 6795-6820.

Digital Library

[16]

A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations. Int. J. Numer. Modell. Electron. Networks Devices Fields. v22. 77-103.

Digital Library

[17]

Dumbser, M., Käser, M. and 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. v171. 695-717.

[18]

Ezziani, A. and Joly, P., Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems. J. Comput. Appl. Math. v234. 1886-1895.

Digital Library

[19]

Constantinescu, E. and Sandu, A., Multirate time-stepping methods for hyperbolic conservation laws. J. Sci. Comput. v33. 239-278.

Digital Library

[20]

Constantinescu, E. and Sandu, A., Multirate explicit Adams methods for time integration of conservation laws. J. Sci. Comput. v38. 229-249.

Digital Library

[21]

Diaz, J. and Grote, M.J., Energy conserving explicit local time-stepping for second-order wave equations. SIAM J. Sci. Comput. v31. 1985-2014.

Digital Library

[22]

Grote, M.J. and Mitkova, T., Explicit local time-stepping for Maxwell's equations. J. Comput. Appl. Math. v234. 3283-3302.

Digital Library

[23]

Hochbruck, M. and Ostermann, A., Exponential multistep methods of Adams-type. BIT. v51. 889-908.

Digital Library

[24]

Lions, J.L. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Vol. I. 1972. Springer-Verlag.

[25]

Cohen, G., High-Order Numerical Methods for Transient Wave Equations. 2002. Springer-Verlag.

[26]

Cohen, G., Joly, P. and Tordjman, N., Construction and analysis of higher order finite elements with mass lumping for wave equation. In: Proceedings of Second International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, SIAM. pp. 152-160.

[27]

Cohen, G., Joly, P. and Tordjman, N., Higher-order finite elements with mass-lumping for the 1D wave equation. Finite Elem. Anal. Des. v16. 329-336.

Digital Library

[28]

Baker, and Douglas, V., The effect of quadrature errors on finite element approximations for second order hyperbolic equations. SIAM J. Numer. Anal. v13. 577-598.

Digital Library

[29]

Buffa, A., Perugia, I. and Warburton, T., The mortar-discontinuous Galerkin method for the 2D Maxwell eigenproblem. J. Sci. Comput. v40. 86-114.

Digital Library

[30]

Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. v39. 1749-1779.

Digital Library

[31]

C. Agut, J. Diaz, Stability analysis of the interior penalty discontinuous Galerkin method for the wave equation, INRIA Research Report 7494, 2010.

[32]

Grote, M.J., Schneebeli, A. and Schötzau, D., Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates. IMA J. Numer. Anal. v28. 440-468.

[33]

Grote, M.J. and Schötzau, D., Optimal error estimates for the fully discrete interior penalty DG method for the wave equation. J. Sci. Comput. v40. 257-272.

Digital Library

[34]

Hairer, E., Nørsett, S. and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems. 2000. Springer.

Digital Library

Cited By

Boscheri WDumbser MZanotti O(2015)High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshesJournal of Computational Physics10.1016/j.jcp.2015.02.052291:C(120-150)Online publication date: 15-Jun-2015
https://dl.acm.org/doi/10.1016/j.jcp.2015.02.052
Demirel ANiegemann JBusch KHochbruck M(2015)Efficient multiple time-stepping algorithms of higher orderJournal of Computational Physics10.1016/j.jcp.2015.01.018285:C(133-148)Online publication date: 15-Mar-2015
https://dl.acm.org/doi/10.1016/j.jcp.2015.01.018
Kostin VLisitsa VReshetova GTcheverda V(2015)Local time-space mesh refinement for simulation of elastic wave propagation in multi-scale mediaJournal of Computational Physics10.1016/j.jcp.2014.10.047281:C(669-689)Online publication date: 15-Jan-2015
https://dl.acm.org/doi/10.1016/j.jcp.2014.10.047
Show More Cited By

High-order explicit local time-stepping methods for damped wave equations
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

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Published In

cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 239, Issue

February, 2013

430 pages

ISSN:0377-0427

Issue’s Table of Contents

Copyright © Elsevier B.V. © 2012.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 February 2013

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Cited By

Boscheri WDumbser MZanotti O(2015)High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshesJournal of Computational Physics10.1016/j.jcp.2015.02.052291:C(120-150)Online publication date: 15-Jun-2015
https://dl.acm.org/doi/10.1016/j.jcp.2015.02.052
Demirel ANiegemann JBusch KHochbruck M(2015)Efficient multiple time-stepping algorithms of higher orderJournal of Computational Physics10.1016/j.jcp.2015.01.018285:C(133-148)Online publication date: 15-Mar-2015
https://dl.acm.org/doi/10.1016/j.jcp.2015.01.018
Kostin VLisitsa VReshetova GTcheverda V(2015)Local time-space mesh refinement for simulation of elastic wave propagation in multi-scale mediaJournal of Computational Physics10.1016/j.jcp.2014.10.047281:C(669-689)Online publication date: 15-Jan-2015
https://dl.acm.org/doi/10.1016/j.jcp.2014.10.047
Guevel YGirault GCadou J(2015)Numerical comparisons of high-order nonlinear solvers for the transient Navier-Stokes equations based on homotopy and perturbation techniquesJournal of Computational and Applied Mathematics10.1016/j.cam.2014.12.008289:C(356-370)Online publication date: 1-Dec-2015
https://dl.acm.org/doi/10.1016/j.cam.2014.12.008
Winters AKopriva D(2014)High-Order Local Time Stepping on Moving DG Spectral Element MeshesJournal of Scientific Computing10.1007/s10915-013-9730-z58:1(176-202)Online publication date: 1-Jan-2014
https://dl.acm.org/doi/10.1007/s10915-013-9730-z

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