Generalized Lax-Friedrichs Schemes for Linear Advection Equation with Damping

To analyze local oscillations existing in the generalized Lax-Friedrichs(LxF) schemes for computing of the linear advection equation with damping, we observed local oscillations in numerical solutions for the discretization of some special initial data under stable conditions. Then we raised three propositions about how to control those oscillations via some numerical examples. In order to further explain this, we first investigated the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceeded to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. We find that the relative phase errors are at least offset by the numerical dissipation of the same order, otherwise the oscillation could be caused. The LxF scheme is conditionally stable and once adding the damping into linear advection equation, the damping has resulted in a slight reduction of the modes' height; We also can find even large damping, the oscillation becomes weaker as time goes by, that is to say the chequerboard mode decay.

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