In this thesis, we investigate four neighboring topics, all in the general area of numerical methods for solving Partial Differential Equations (PDEs).
Topic 1. Radial Basis Functions (RBF) are widely used for multi-dimensional interpolation of scattered data. This methodology offers smooth and accurate interpolants, which can be further refined, if necessary, by clustering nodes in select areas. We show, however, that local refinements with RBF (in a constant shape parameter ϵ regime) may lead to the oscillatory errors associated with the Runge phenomenon (RP). RP is best known in the case of high-order polynomial interpolation, where its effects can be accurately predicted via Lebesgue constant Λ (which is based solely on the node distribution). We study the RP and the applicability of Lebesgue constant (as well as other error measures) in RBF interpolation. Mainly, we allow for a spatially variable shape parameter, and demonstrate how it can be used to suppress RP-like edge effects and to improve the overall stability and accuracy.
Topic 2. Although not as versatile as RBFs, cubic splines are useful for interpolating grid-based data. In 2-D, we consider a patch representation via Hermite basis functions si,j( u, v) = [special characters omitted] hmnHm(u) Hn(v), as opposed to the standard bicubic representation. Stitching requirements for the rectangular non-equispaced grid yield a 2-D tridiagonal linear system AX = B, where X represents the unknown first derivatives. We discover that the standard methods for solving this N×M system do not take advantage of the spline-specific format of the matrix B. We develop an alternative approach using this specialization of the RHS, which allows us to pre-compute coefficients only once, instead of N times. M
ATLAB
implementation of our fast 2-D cubic spline algorithm is provided. We confirm analytically and numerically that for large
N (
N > 200), our method is at least 3 times faster than the standard algorithm and is just as accurate.
Topic 3. The well-known ADI-FDTD method for solving Maxwell's curl equations is second-order accurate in space/time, unconditionally stable, and computationally efficient. We research Richardson extrapolation-based techniques to improve time discretization accuracy for spatially oversampled ADI-FDTD. A careful analysis of temporal accuracy, computational efficiency, and the algorithm's overall stability is presented. Given the context of wave-type PDEs, we find that only a limited number of extrapolations to the ADI-FDTD method are beneficial, if its unconditional stability is to be preserved. We propose a practical approach for choosing the size of a time step that can be used to improve the efficiency of the ADI-FDTD algorithm, while maintaining its accuracy and stability.
Topic 4. Shock waves and their energy dissipation properties are critical to understanding the dynamics controlling the MHD turbulence. Numerical advection algorithms used in MHD solvers (e.g. the ZEUS package) introduce undesirable numerical viscosity. To counteract its effects and to resolve shocks numerically, Richtmyer and von Neumann's artificial viscosity is commonly added to the model. We study shock power by analyzing the influence of both artificial and numerical viscosity on energy decay rates. Also, we analytically characterize the numerical diffusivity of various advection algorithms by quantifying their diffusion coefficients η.*
*This dissertation is a compound document (contains both a paper copy and a CD as part of the dissertation). The CD requires the following system requirements: Adobe Acrobat.
