Practical aspects of higher-order numerical schemes for wave propagation phenomena

This paper examines practical issues related to the use of compact-difference-based fourth- and sixth-order schemes for wave propagation phenomena with focus on Maxwell's equations of electromagnetics. An outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches: transformed plane method and physical space differencing. In the first technique, the truncation error expansion for the sixth-order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available. Physical space-differencing formulas are derived for the five-point stencil by expressing the coefficients in terms of local spacing ratios. The order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes. To ensure stability for a broad range of problems, Fourier analysis is employed to develop a single-parameter family of up to tenth-order tridiagonal-based spatial filters. The implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1-D cavity where they are employed to suppress a late-time instability. The paper concludes after demonstrating the application of the scheme to several 3-D canonical problems utilizing Cartesian as well as curvilinear meshes. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. Government work and is in the public domain in the United States.