Abstract
The occurrence of oscillations in a well-known asymptotic preserving(AP) numerical scheme is investigated in the context of a linearmodel of diffusive relaxation, known as the $P_1$ equations. Thescheme is derived with operator splitting methods that separate the$P_1$ system into slow and fast dynamics. A careful analysis of thescheme shows that binary oscillations can occur as a result of a black-reddiffusion stencil and that dispersive-type oscillations may occurwhen there is too little numerical dissipation. The latterconclusion is based on comparison with a modified form of the $P_1$system. Numerical fixes are also introduced to remove theoscillatory behavior.
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