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An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers’ equation

Rottmann-Matthes, Jens

Abstract: In this paper we introduce a new, simple and efficient numerical scheme for the implementation of the freezing method for capturing similarity solutions in partial differential equations. The scheme is based on an IMEX-Runge-Kutta approach for a method of lines (semi-)discretization of the freezing partial differential algebraic equation (PDAE). We prove second order convergence for the time discretization at smooth solutions in the ODE-sense and we present numerical experiments that show second order convergence for the full discretization of the PDAE. As an example serves the multi-dimensional Burgers’ equation. By considering very different sizes of viscosity, Burgers’ equation can be considered as a prototypical example of general coupled hyperbolicparabolic PDEs. Numerical experiments show that our method works perfectly well for all sizes of viscosity, suggesting that the scheme is indeed suitable for capturing similarity solutions in general hyperbolic-parabolic PDEs by direct forward simulation with the freezing method.

Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2016
Sprache Englisch
Identifikator DOI(KIT): 10.5445/IR/1000063411
ISSN: 2365-662X
URN: urn:nbn:de:swb:90-634117
KITopen ID: 1000063411
Verlag KIT, Karlsruhe
Umfang 26 S.
Serie CRC 1173 ; 2016/38
Schlagworte similarity solutions, relative equilibria, Burgers’ equation, freezing method, scaling symmetry, IMEX-Runge-Kutta, central scheme, hyperbolic-parabolic partial differential algebraic equations
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