Abstract
Grid staggering for wave equations is a validated approach for many applications, as it generally enhances stability and accuracy. This paper is about time staggering. Our aim is to assess a fourth-order, explicit, time-staggered integration method from the literature, through a comparison with two alternative fourth-order, explicit methods. These are the classical Runge-Kutta method and a symmetric-composition method derived from symplectic Euler.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abarbanel, S., Gottlieb, D., Carpenter, M.H.: On the removal of boundary errors caused by Runge-Kutta integration of nonlinear partial differential equations. SIAM J. Sci. Comput. 17, 777–782 (1996)
Blanes, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. J. Comput. Appl. Math. 142, 313–330 (2002)
Calvo, M.P., Palencia, C.: Avoiding the order reduction of Runge-Kutta methods for linear initial-boundary value problems. Math. Comput. 71, 1529–1543 (2002)
Ghrist, M.L.: High-order finite difference methods for wave equations. Ph.D. Thesis, University of Colorado, Boulder, USA (2000)
Ghrist, M., Fornberg, B., Driscoll, T.A.: Staggered time integrations for wave equations. SIAM. J. Numer. Anal. 38, 718–741 (2000)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol. 31. Springer, New York (2002)
Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, New York (2003)
Kinnmark, I.P.E., Gray, W.G.: One-step integration methods of third-fourth order accuracy with large hyperbolic stability limits. Math. Comput. Simul. 18, 181–184 (1984)
McLachlan, R.I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16, 151–168 (1995)
McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341–434 (2002)
Murua, A., Sanz-Serna, J.M.: Order conditions for numerical integrators obtained by composing simpler integrators. Philos. Trans. Roy. Soc. A 357, 1079–1100 (1999)
Pathria, D.: The correct formulation of intermediate boundary conditions for Runge-Kutta time integration of initial boundary value problems. SIAM J. Sci. Comput. 18, 1255–1266 (1997)
Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)
Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50, 405–418 (1987)
Sonneveld, P., van Leer, B.: A minimax problem along the imaginary axis. Nieuw Archief Wiskd. 3, 19–22 (1985)
van der Houwen, P.J.: Construction of Integration Formulas for Initial Value Problems. North-Holland, Amsterdam (1977)
Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302 (1966)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Verwer, J.G. On Time Staggering for Wave Equations. J Sci Comput 33, 139–154 (2007). https://doi.org/10.1007/s10915-007-9146-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-007-9146-8