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December  2008, 1(4): 573-590. doi: 10.3934/krm.2008.1.573

Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation

Jeffrey R. Haack 1, and  Cory D. Hauck 2,

1. 

Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706, United States
2. 

Computational Physics Group (CCS-2) and Center for Nonlinear Studies (T-CNLS), Mail Stop B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, United States

Received  September 2008 Revised  September 2008 Published  October 2008

The occurrence of oscillations in a well-known asymptotic preserving (AP) numerical scheme is investigated in the context of a linear model of diffusive relaxation, known as the $P_1$ equations. The scheme is derived with operator splitting methods that separate the $P_1$ system into slow and fast dynamics. A careful analysis of the scheme shows that binary oscillations can occur as a result of a black-red diffusion stencil and that dispersive-type oscillations may occur when there is too little numerical dissipation. The latter conclusion is based on comparison with a modified form of the $P_1$ system. Numerical fixes are also introduced to remove the oscillatory behavior.
Keywords: splitting, hyperbolic heat equation, diffusive relaxation, Asymptotic-preserving scheme.
Mathematics Subject Classification: 65M06, 82C70, 82C80.
Citation: Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic and Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573
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