Using time filtering to control the long-time instability in seismic wave simulation

Long-time instabilities have been observed in various scenarios of numerical simulation for seismic wave propagation. They appear as slowly magnifying spurious oscillations in the seismograms at the late stage of the simulation. Their magnifying speed is typically much slower than the magnifying speed observed when the Courant-Friedrichs-Lewy condition is violated. The simulations can therefore continue to proceed for a relatively long period without floating-point overflow. The impact of the long-time instabilities on the simulation accuracy at the early stage can be negligible in some cases. In existing literatures, spatial-filtering techniques that, in principle, average the solution within certain spatial range at the same time level are typically utilized to control the long-time instability. In this paper, we present an alternative time-filtering approach that, in principle, averages the solution at different time levels of the same spatial location to control the long-time instability. Comparing with the spatial filtering, the advantages of this time-filtering approach lie in its flexibility, particularly when boundaries or interfaces are involved, its simplicity and low additional arithmetic operations, at the expense of extra memory cost. When application of the time filtering is localized to regions where long-time instabilities are emitted from, for example, a boundary or an interface layer, the additional cost is negligible when compared with the cost of wave simulation. For linear wave equations, this time-filtering approach can be understood as the introduction of artificial diffusion. Its application has impact on the accuracy of the solution and the restriction of the time step size. We present an indicator-based approach to adjust the filtering parameters both spatially and temporally, in order to provide the best trade-off between accuracy and stability. The indicator is calculated heuristically by monitoring the spurious oscillation as the simulation evolves in time.