Higher Order Time Stepping for Second Order Hyperbolic Problems and Optimal CFL Conditions
We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail and the analysis results in a specific numerical algorithm. The corresponding results are quite promising and suggest various conjectures.
KeywordsChebyshev Polynomial Equation Approach Tangent Point Order Scheme Optimal Polynomial
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- Che66.E. W. Cheney. Introduction to Approximation Theory. McGraw-Hill, 1966.Google Scholar
- Coh02.G. C. Cohen. Higher-order numerical methods for transient wave equations. Scientific Computation. Springer-Verlag, Berlin, 2002.Google Scholar
- Deu04.P. Deuflhard. Newton Methods for Nonlinear Problems – Affine Invariance and Adaptative Algorithms. Number 35 in Computational Mathematics. Springer, Berlin, 2004.Google Scholar
- HW96.E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd edition, 1996. Stiff and differential-algebraic problems.Google Scholar
- Jol03.P. Joly. Variational methods for time-dependent wave propagation problems. In Topics in computational wave propagation, volume 31 of Lect. Notes Comput. Sci. Eng., pages 201–264. Springer, Berlin, 2003.Google Scholar
- Kan01.Ch. Kanzow. An active set-type Newton method for constrained nonlinear systems. In M.C. Ferris, O.L. Mangasarian, and J.S. Pang, editors, Complementarity: applications, algorithms and extensions, pages 179–200, Dordrecht, 2001. Kluwer Acad. Publ.Google Scholar
- RM67.R. D. Richtmyer and K. W. Morton. Difference methods for initial-value problems, volume 4 of Interscience Tracts in Pure and Applied Mathematics. John Wiley & Sons, Inc., New York, 2nd edition, 1967.Google Scholar
- RS78.M. Reed and B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.Google Scholar
- Wei06.E. W. Weisstein. Chebyshev polynomial of the first kind. MathWorld. http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html , 2006.