Tridiagonal matrix algorithm

Abstract: Described here are the design and implementation of a family of algorithms for a variety of classes of narrowly banded linear systems. The classes of matrices include symmetric and positive de nite, nonsymmetric but diagonally dominant, and general nonsymmetric; and, all these types are addressed for both general band and tridiagonal matrices. The family of algorithms captures the general avor of existing divide-and-conquer algorithms for banded matrices in that they have three distinct phases, the rst and last of which are completely parallel, and the second of which is the parallel bottleneck. The algorithms have been modi ed so that they have the desirable property that they are the same mathematically as existing factorizations (Cholesky, Gaussian elimination) of suitably reordered matrices. This approach represents a departure in the nonsymmetric case from existing methods, but has the practical bene ts of a smaller and more easily handled reduced system. All codes implement a block odd-even reduction for the reduced system that allows the algorithm to scale far better than existing codes that use variants of sequential solution methods for the reduced system. A cross section of results is displayed that supports the predicted performance results for the algorithms. Comparison with existing dense-type methods shows that for areas of the problem parameter space with low bandwidth and/or high number of processors, the family of algorithms described here is superior.

Abstract: To solve tridiagonal systems of linear equations, the Thomas Algorithm is a much more efficient method than, for instance, Gaussian elimination. The algorithm uses a series of elementary row operations and can solve a system of n equations in (n) operations, instead of (n3) . Many variations of the Thomas Algorithm have been developed over the years to solve very specific near-tridiagonal matrix. However, none of these methods address the situation of a system of linear equations that could easily be solved if elementary operations on columns are applied, instead of elementary operations on rows. The present paper proposes an efficient method that allows the use of elementary column operations to solve linear systems of equations using vector multiplication techniques, such as the one proposed by Thomas. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract: This paper treats the mathematical derivation of a novel formulation of the Navier–Stokes equation for general non-orthogonal curvilinear co-ordinates. The covariant velocity components are solved in this FVM formulation, which leads to the pressure-velocity coupling becoming relatively easy to handle at the expense of a more complicated expression of the convective and diffusive fluxes. When a velocity component is solved at a point P, the neighbouring velocities are projected in the direction of the velocity component at the point P. Thus the base vectors are changed at the neighbouring points. This renders a simpler expression for the covariant derivatives. Neither the Cristoffel symbol nor its derivatives need be computed. This contributes to the accuracy of the formulation. The procedure of changing the base vectors affects only the convected velocity. The convecting term (dot product of velocity and area) is calculated without any change of the base vectors. The same is true for the operator on the covariant velocity in the diffusion term. It is shown that when using upwind differencing the use of projected velocities gives better results than when curvature effects are included in the source term. The discretized equations are written in a form which enables the use of the tridiagonal matrix algorithm (TDMA). The equations can be solved using either the SIMPLEC or the PISO procedure. Two examples of laminar flows are given.

Abstract: We develop a finite difference method of the Crank?Nicholson type for solving three-dimensional heat transport equations in a double-layered thin film with microscale thickness. The three-dimensional implicit scheme is solved by using a preconditioned Richardson iteration, so that only two tridiagonal linear systems with unknowns at the interface are solved at each iteration. We then apply a parallel Gaussian elimination to solve these two tridiagonal linear systems and develop a domain decomposition algorithm for thermal analysis of the double-layered thin film. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained.