# American Institute of Mathematical Sciences

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

 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.
Citation: Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573
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