Time Filtered Finite Difference Schemes for Linear Hyperbolic Problems

The focus of this paper is to investigate the effects of combining a time filter with three finite difference methods for numerically solving a linear hyperbolic equation. The examples in this paper show that, in some cases, the filter technique can be successfully combined with an explicit scheme. The three examples demonstrate different possible outcomes when one attempts to apply filtering. Von Neumann analysis proves to be a useful tool for examining the stability properties of filtered versions of the upwind and leapfrog methods as it allows one to deal with the filter parameter systematically. In each case, the analysis of the filtered scheme is done with insight into that of the original un-filtered method. With a careful choice of the filter parameter, the filtered upwind scheme is shown to be more accurate than its upwind counterpart, but the analysis also shows that a new CFL condition must be used in order to obtain accurate, stable results. This paper also demonstrates that the type of filter considered here cannot remedy certain kinds of stability properties. The filtered leapfrog scheme is shown to inherit the same type of spurious mode that its original un-filtered counterpart is known to possess. We also show that combining the filter with the Crank-Nicolson method leads to an implicit method where no value of the filter parameter provides for a consistent filtered version of the method. We conclude with numerical computations that support the theoretical results for the improved accuracy of the filtered upwind scheme.