A Comprehensive New Methodology for Formulating FDTD Schemes With Controlled Order of Accuracy and Dispersion

Abstract

Numerical dispersion errors in the wave-equation-finite-difference-time-domain (WE-FDTD) method have been treated by higher order schemes, coefficient modification schemes, dispersion relation preserving and non-standard schemes. In this work, a unified methodology is formulated for the systematic generation of WE-FDTD schemes tailored to the spectrum of the excitation. The methodology enables the scheme designer to gradually trade order of accuracy (OoA) for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior, that is not deemed critical for this type of excitation. The methodology is shown to encompass both existing and new schemes. Stability analysis is carried out concurrently with the generation of each scheme. Using a stencil size of 3 and 5 temporal and spatial samples, respectively, long term errors of a scheme designed for a specific pulse are compared with the standard (4,4) scheme that has the same computational complexity, via simulation of a modulated pulse that propagates over a million time steps.Index Terms-Finite-difference-time-domain (FDTD), numerical dispersion, order of accuracy (OoA), wave equation.

The spectral order of accuracy: A new unified tool in the design methodology of excitation-adaptive wave equation FDTD schemes

“…broadband dispersion error, anisotropy) methods have been developed, e.g. in [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”

A new time–space domain high-order finite-difference method for the acoustic wave equation

“…We have developed a new FD method in time-space domain for the acoustic wave equation based on the methods in [39,40]; the FD coefficients are determined by the Courant number and space point number. The new method has greater accuracy than the conventional method under the same discretization.…”

“…However, the details of our approach are different in that we derive coefficients that are better suited to modeling in time domain comprising a continuous spectrum of frequencies. In our paper, the spatial and temporal derivatives are treated separately first, and then combined using the wave equation, whereas in [40] both derivatives for the 1D wave equation are approximated concurrently while already embedded in the equation.…”

“…Since the source spectrum contains a continuous band of frequencies, our method determines the FD coefficients by using the Taylor series expansion in the dispersion relation. Our FD coefficients of 1D are, however, equivalent to those in [40]. The details of our derivation are outlined below.…”

Scalar Wave Equation Modeling with Time-Space Domain Dispersion-Relation-Based Staggered-Grid Finite-Difference Schemes

“…The key idea of this method is that the dispersion relation is completely satisfied at designated frequencies; thus several equations are formed, and the FD coefficients are obtained by solving these equations. This method was developed further for the one-dimensional 1D lossless and boundless wave equation, and its spatial FD coefficients were determined at one designated frequency to obtain arbitraryorder accuracy (Finkelstein and Kastner, 2008). Liu and Sen (2009c) employed a plane wave theory and the Taylor series expansion of dispersion relation to derive the FD coefficients in the joint time-space domain for the scalar wave equation with second-order spatial derivatives.…”