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Practical aspects of higher-order numerical schemes for wave propagation phenomena. (English) Zbl 0959.65103
Summary: 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 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.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
Full Text: DOI
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