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

Abstract

This paper examines practical issues related to the use of compact-di erence-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 di erencing. In the ÿrst technique, the truncation error expansion for the sixth-order compact scheme conÿrms that the order of accuracy is preserved if a consistent mesh reÿnement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available. Physical space-di erencing formulas are derived for the ÿve-point stencil by expressing the coe cients in terms of local spacing ratios. The order of accuracy of the reconstruction operator is then veriÿed 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 ÿlters. The implementation of these ÿlters is discussed in terms of their e ect 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.

A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries

“…These boundary schemes are discussed and suggested in Ref.[12]. Since a central compact scheme has no numerical dissipation, an implicit spatial filtering proposed by Gaitonde et al[41] is applied to suppress aliasing errors and ensure numerical stability. A general M -th order spatial filtering formulation is written as…”

“…where −0.5 < α f < 0.5 is a free parameter, and coefficients a n can be found in Ref.[41]. For a value of α f which is close to 0.5, the filtering truncates only very high wave number components.…”

Simulation of three‐dimensional incompressible flows in generalized curvilinear coordinates using a high‐order compact finite‐difference lattice Boltzmann method

“…Note that the distribution function f α at the end points, i = 1, Imax , j = 1, Jmax , and k = 1, kmax , are not filtered. At other near boundary points where Equation cannot be used, higher‐order one‐sided formulas proposed in the work of Gaitonde et al is applied to retain the tridiagonal form of the filter scheme. In the present calculations, the filter is applied to the distribution function f α , sequentially in the ξ −, η − and ζ −coordinate directions, once after each time step of the algorithm.…”

Fundamentals and Applications of Modern Flow Control

“…As discussed by Gaitonde et al (1999), the order of the fi lter can have a signifi cant impact: lowering the order provides better fi ltering at the spurious frequencies, but at the cost of reducing the range of resolvable frequencies. Many methods seem to be successfully employed, including second-order central differencing, higherorder compact differencing, and pseudo-spectral/spectral methods.…”

“…The charge density is assumed to be uniform in the region of plasma formation. Spatial derivatives of the RANS equations were approximated with a sixth-order compact differencing method (Gaitonde et al, 1999;Visbal and Gaitonde, 2001). As examples of computations involving fl ow control with plasma actuators, we consider fl ow over a turbine blade, wall-mounted hump model, and a wing section.…”