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Register a new userA second mapping method in generalized discrete singular convolution algorithm: regularizing singularities for one electron system
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A second mapping method is introduced in the generalized discrete singular convolution algorithm. The mapping approaches are adopted to regularize singularities for one electron system. The applications of the two mapping methods are generalized from the radial hydrogen problem to the one-dimensional hydrogen problem. Three mapping functions are chosen: the square-root mapping function, the cube-root mapping function, and the logarithm mapping function. However, the present mapping approaches fail in both the two-dimensional and three-dimensional hydrogen problems, because the wavefunctions of s-states at the nuclei are not correct.
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In this paper, authors focus effort on improving the conventional discrete velocity method (DVM) into a multiscale scheme in finite volume framework for gas flow in all flow regimes. Unlike the typical multiscale kinetic methods unified gas-kinetic scheme (UGKS) and discrete unified gas-kinetic scheme (DUGKS), which concentrate on the evolution of the distribution function at the cell interface, in the present scheme the flux for macroscopic variables is split into the equilibrium part and the nonequilibrium part, and the nonequilibrium flux is calculated by integrating the discrete distribution function at the cell center, which overcomes the excess numerical dissipation of the conventional DVM in the continuum flow regime. Afterwards, the macroscopic variables are finally updated by simply integrating the discrete distribution function at the cell center, or by a blend of the increments based on the macroscopic and the microscopic systems, and the multiscale property is achieved. Several test cases, involving unsteady, steady, high speed, low speed gas flows in all flow regimes, have been performed, demonstrating the good performance of the multiscale DVM from free molecule to continuum Navier-Stokes solutions and the multiscale property of the scheme is proved.
We propose a harmonic surface mapping algorithm (HSMA) for electrostatic pairwise sums of an infinite number of image charges. The images are induced by point sources within a box due to a specific boundary condition which can be non-periodic. The HSMA first introduces an auxiliary surface such that the contribution of images outside the surface can be approximated by the least-squares method using spherical harmonics as basis functions. The so-called harmonic surface mapping is the procedure to transform the approximate solution into a surface charge and a surface dipole over the auxiliary surface, which becomes point images by using numerical integration. The mapping procedure is independent of the number of the sources and is considered to have a low complexity. The electrostatic interactions are then among those charges within the surface and at the integration points, which are all the form of Coulomb potential and can be accelerated straightforwardly by the fast multipole method to achieve linear scaling. Numerical calculations of the Madelung constant of a crystalline lattice, electrostatic energy of ions in a metallic cavity, and the time performance for large-scale systems show that the HSMA is accurate and fast, and thus is attractive for many applications.
A generalized eigenvalue algorithm for tridiagonal matrix pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil. The convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system.
We have carried out an analysis of singularities in Kohn variational calculations for low energy e^{+}-H_{2} elastic scattering. Provided that a sufficiently accurate trial wavefunction is used, we argue that our implementation of the Kohn variational principle necessarily gives rise to singularities which are not spurious. We propose two approaches for optimizing a free parameter of the trial wavefunction in order to avoid anomalous behaviour in scattering phase shift calculations, the first of which is based on the existence of such singularities. The second approach is a more conventional optimization of the generalized Kohn method. Close agreement is observed between the results of the two optimization schemes; further, they give results which are seen to be effectively equivalent to those obtained with the complex Kohn method. The advantage of the first optimization scheme is that it does not require an explicit solution of the Kohn equations to be found. We give examples of anomalies which cannot be avoided using either optimization scheme but show that it is possible to avoid these anomalies by considering variations in the nonlinear parameters of the trial function.
Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a finite domain. Restricting the problem to a finite domain introduces a localization error and it is proved that this localization error stays below a given threshold if the finite domain is large enough. For two particular kernel functions, the numerical examples concerning solitary wave solutions illustrate the expected accuracy of the method. Our class of nonlocal wave equations includes the Benjamin-Bona-Mahony equation as a special case and the present work is inspired by the previous work of Bona, Pritchard and Scott on numerical solution of the Benjamin-Bona-Mahony equation.
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