Some new approximation methods are described for harmonic functions corresponding to boundary values on the (unit) sphere. Starting from the usual Fourier (orthogonal) series approach, we propose here nonorthogonal expansions, i.e. series expansions in terms of overcomplete systems consisting of localizing functions. In detail, we are concerned with the so-called Gabor, Toeplitz, and wavelet expansions. Essential tools are modulations, rotations, and dilations of a mother wavelet. The Abel-Poisson kernel turns out to be the appropriate mother wavelet in approximation of harmonic functions from potential values on a spherical boundary.
In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction of the discretization parameter as well as on the differential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit particle scheme for the Boltzmann equation with Maxwellian molecules. The present paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second order particle method, when using an equiweighting of explicit and implicit discretization.
The paper presents a numerical simulation technique - based on the well-known particle methods - for the stationary, one-dimensional Boltzmann equation for Maxwellian molecules. In contrast to the standard splitting methods, where one works with the instationary equation, the current approach simulates the direct solution of the stationary problem. The model problem investigated is the heat transfer between two parallel plates in the rarefied gas regime. An iteration process is introduced which leads to the stationary solution of the exact - space discretized - Boltzmann equation, in the sense of weak convergence.
Based on state-space formulas for coprime factorizations over ... and an algebraic characterization of J-inner functions, normalized doubly-coprime factorizations for different classes of continuous- and discrete-time transfer functions are derived by using a single general construction method. The parametrization of the factors is in terms of the stabilizing solutions of general degenerate continuous- respectively discrete-time Riccati equations, which are obtained by examining state-space representations of J-normalized factor matrices.
This paper deals with domain decomposition methods for kinetic and drift diffusion semiconductor equations. In particular accurate coupling conditions at the interface between the kinetic and drift diffusion domain are given. The cases of slight and strong nonequilibrium situations at the interface are considered and some numerical examples are shown.
A way to derive consistently kinetic models for vehicular traffic from microscopic follow the leader models is presented. The obtained class of kinetic equations is investigated. Explicit examples for kinetic models are developed with a particular emphasis on obtaining models, that give realistic results. For space homogeneous traffic flow situations numerical examples are given including stationary distributions and fundamental diagrams.
By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show now spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.
Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper, that even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results ofXu and Cheney.
A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed 2-dimensional sequences, rotations on the sphere, triangulation, and sum of three squares sequence, are investigated. Quantitative tests are done, and the results are compared with each other. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.