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On leapfrog-Chebyshev schemes

Carle, Constantin; Hochbruck, Marlis; Sturm, Andreas

This paper is dedicated to the improvement of the efficiency of the leapfrog method for linear and semilinear second-order differential equations. In numerous situations the strict CFL condition of the leapfrog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have been constructed for linear problems that exhibit a much weaker CFL condition than the leapfrog method (at a higher cost). However, these methods fail to produce the correct long-time behavior of the exact solution which can result in a bad approximation quality.
In this paper we introduce a new class of leapfrog-Chebyshev methods for semilinear problems. For the linear part, we use Chebyshev polynomials while the nonlinearity is treated by the standard leapfrog method. The method can be viewed as a multirate scheme because the nonlinearity is evaluated only once in each time step whereas the number of evaluations of the linear part corresponds to the degree of the Chebyshev polynomial. In contrast to existing literature (which is restricted to linear problems), we suggest to stabilize the scheme and we introduce a new starting value required for the two-step method.
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DOI: 10.5445/IR/1000099118
Veröffentlicht am 21.10.2019
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2019
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000099118
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 29 S.
Serie CRC 1173 ; 2019/19
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Schlagwörter time integration, Hamiltonian systems, wave equation, second-order ode, leapfrog method, CFL condition, Chebyshev polynomials, stability analysis, error analysis, generating functions
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