Browsing by Subject "Asymptotic expansions"
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Item Asymptotic analysis of the spatial weights of the arbitrarily high order transport method(2001-08) Elsawi, Mohamed Abdel Halim; Abdurrahman, Naeem M.; Koen, B. V.We perform an asymptotic analysis to the spatial weights of the Arbitrarily High Order Transport (AHOT) method that employs the method of characteristic as a means to relate the surface and the average fluxes of a computational cell in twodimensional Cartesian geometry. Previously, the spatial weights of the AHOT’s final discrete-variable equations have shown some numerical instabilities as the cell optical thickness approaches zero (or when computed on sufficiently thin cells). Our analysis is based on identifying the components of the spatial weights that are responsible for these instabilities, then expanding them in a truncated power series of the cell optical thickness that causes the instabilities when approaches zero. We then derive the conditions necessary to eliminate the singularity as the cell optical thickness approaches zero. We show that the method we adopted for computing the asymptotic spatial weights is very effective in eliminating these numerical instabilities by comparing the weights using the full analytical expressions and the new asymptotic ones, both being computed on fine grids. We implement the new asymptotic formulas for the spatial weights in the AHOT-C test code and generate benchmark quality solutions to some of the Burre’s Suite of Test Problems (BSTeP), which can be used as reference solutions to verify the accuracy of other solution methods and algorithms.Item Quadrature, interpolation and observability(Texas Tech University, 1997-12) Hodges, Lucille McDanielMethods of interpolation and quadrature have been used for over 300 years. Improvements in the techniques have been made by many, most notably by Gauss, whose technique applied to polynomials is referred to as Gaussian Quadrature. Stieltjes extended Gauss's method to certain non-polynomial functions as early as 1884. Conditions that guarantee the existence of quadrature formulas for certain collections of functions were studied by TchebychefF, and his work was extended by others. Today, a class of functions which satisfies these conditions is called a Tchebycheff System. This thesis contains the definition of a TchebychefF System, along with the theorems, proofs, and definitions necessary to guarantee the existence of quadrature formulas for such systems. Solutions of discretely observable linear control systems are of particular interest, and observability with respect to a given output function is defined. The output function is written as a linear combination of a collection of orthonormal functions. Orthonormal functions are defined, and their properties are discussed. The technique for evaluating the coefficients in the output function involves evaluating the definite integral of functions which can be shown to form a Tchebycheff system. Therefore, quadrature formulas for these integrals exist, and in many cases are known. The technique given is useful in cases where the method of direct calculation is unstable. The condition number of a matrix is defined and shown to be an indication of the the degree to which perturbations in data affect the accuracy of the solution. In special cases, the number of data points required for direct calculation is the same as the number required by the method presented in this thesis. But the method is shown to require more data points in other cases. A lower bound for the number of data points required is given.Item Reaction-diffusion fronts in inhomogeneous media(2006) Nolen, James Hilton; Xin, Jack; Souganidis, PanagiotisIn this thesis, we study the asymptotic behavior of solutions to the reaction-advection-diffusion equation ut = ∆zu + B(z, t) · ∇zu + f(u), z ∈ R n , t > 0 under various conditions on the prescribed flow B. Our goal is to characterize, bound, and compute the speed of propagating fronts that develop in the solution u and to describe their dependence on the flow B. We focus mainly on the case when f is the KPP nonlinearity f(u) = u(1 − u). In the first section, we consider the case that B is a temporally random field having a spatial shear structure and Gaussian statistics. We show that the solution to the initial value problem develops traveling fronts, almost surely, which are characterized by a deterministic variational principle. In the second section, we use this and other variational principles to derive analytical estimates on the speed of propagating fronts. In the final section, we use the variational principle to compute the front speed numerically. The mathematical analysis involves perturbation expansions, ergodic theorems, and techniques from the theory of large deviations. We use numerical methods for computing the principal Lyapunov exponents of parabolic operators, which appear in the variational characterization of the front speed.