Tridiagonal matrix algorithm

Abstract: A method for the inversion of block tridiagonal matrices encountered in electronic structure calculations is developed, with the goal of efficiently determining the matrices involved in the Fisher–Lee relation for the calculation of electron transmission coefficients. The new method leads to faster transmission calculations compared to traditional methods, as well as freedom in choosing alternate Green’s function matrix blocks for transmission calculations. The new method also lends itself to calculation of the tridiagonal part of the Green’s function matrix. The effect of inaccuracies in the electrode self-energies on the transmission coefficient is analyzed and reveals that the new algorithm is potentially more stable towards such inaccuracies. 2007 Elsevier Inc. All rights reserved. PACS: 71.15. m; 02.70. c

Abstract: Efficient Gaussian elimination method for symbolic determinants and linear systems

Abstract: This chapter presents some of the basic computational techniques that can be employed to solve the governing equations of fluid dynamics. The first stage of obtaining the computational solution involves the conversion of the governing equations into a system of algebraic equations. This is usually known as the discretization stage. It discusses some of the discretization tools, such as the finite- difference and finite-volume methods, which form the foundation of understanding the basic features of discretization. Both of these methods are abundant in many CFD applications. The second stage involves numerically solving the system of algebraic equations, which can be achieved by either the direct methods or iterative methods. Basic direct methods such as the Gaussian elimination and the Thomas algorithm are discussed. Simple iterative methods such as the point-by-point Jacobi and Gauss-Siedel methods are also described. Nevertheless, CFD problems are generally multidimensional and comprise a large system of equations to be solved. Efficient iterative methods such as the ADI or Stone's SIP are often applied to solve such a system of equations. To further enhance the convergence of the computational solution, precondition conjugate gradient methods or multigrid methods are employed to accelerate the iteration process. Finally, this chapter discusses the assessment of convergence. In practice, the algebraic equations that result from the discretization process yield the flow solution at each nodal point on a finite-grid layout.

Abstract: This paper presents the mathematical model of the thermal power plant in cooling pond under different hydrometeorological conditions, which is solved by three dimensional Navier - Stokes equations and temperature equation for an incompressible fluid in a stratified medium. A numerical method based on the projection method, which divides the problem into three stages. At the first stage it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by method of fractional steps. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). Finally at the third stage it is expected that the transfer is only due to the pressure gradient. To increase the order of approximation compact scheme was used. Then qualitatively and quantitatively approximate the basic laws of the hydrothermal processes depending on different hydrometeorological conditions are determined.