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: A heterogeneous fluid dynamic model has been developed to describe the complex flow structure of two-phase flow in bubble columns. The equation of continuity and the momentum balances are the basis of the model. The coupling of the two phases is performed by a force of interaction which is deduced by a force balance around a single rising bubble. Multiphase flow mixing processes are taken into consideration by introducing the turbulent viscosities of the two phases involved. The model equations were implemented successfully by applying a tridiagonal matrix algorithm.

Abstract: This report presents several ideas for improving the stability of Gaussian reduction of an arbitrary real matrix to tridiagonal form. First, we analyze conditions under which reduction algorithms break down or become unstable. Second, we discuss how methods of threshold pivoting decrease the probability of these conditions occurring. Finally, we present new methods for recovering from breakdown when it does occur. The class of matrices that can be successfully reduced is significantly broadened by these new recovery algorithms. 16 refs., 4 figs.

Abstract: Gaussian elimination with partial pivoting is the standard numerical algorithm for solving unstructured linear systems. Here it is shown that Gaussian elimination with partial pivoting or complete pivoting is log-space complete for P. This provides theoretical evidence that these algorithms cannot be efficiently implemented on a highly parallel computer with a large number of processors. Since other algorithms for linear systems that are efficient on parallel computers are already known, this suggests that elimination-based approaches should not be pursued in a parallel environment with many processors.

Abstract: A new theorem giving a direct solution for the inverse or a block tridiagonal matrix is proved. The algorithm is compared to that of Thomas and found to be less efficient in the general case, but more efficient in several practical special cases. In particular the algorithm is shown to be the general case of the various Marching Algorithms whose application to distributed systems is well known.

Abstract: A new method for the solution of pentadiagonal systems of linear equations is presented. The method is a generalization of ordinary odd-even elimination used for tridiagonal systems. Using n processors, an n X n pentadiagonal system can be solved using the new method (generalized odd-even elimination) in time proportional to log2n.

Abstract: Many algorithms for solving the Navier-Stokes equations require the solution of periodic block tridiagonal systems of equations. By applying a splitting to the matrix representing this system of equations, it may first be reduced to a block tridiagonal matrix plus an outer product of two block vectors. The Sherman-Morrison-Woodbury formula is then applied. The algorithm thus reduces a periodic banded system to a non-periodic banded system with additional right-hand sides and is of higher efficiency than standard Thomas algorithm/LU decompositions.

Abstract: We present a fast vector algorithm which solves tridiagonal linear equations by an optimum synthesis of the inherently recursive Gaussian elimination and the parallel though complex cyclic reduction. The idea is to perform an incomplete cyclic reduction to bring the dimension of the tridiagonal system efficiently below a characteristic size n ∗ and then to solve the remaining system by Gaussian elimination. Extensive numerical experiments on the CYBER 205 and the CRAY X-MP computers reveal a maximum vector speedup of 13 and prove n ∗ to reflect the architecture of the vector computer. The performance is further enhanced when a feq right-hand sides are treated simultaneously.

Abstract: The recursive doubling algorithm as developed by Stone can be used to solve a tridiagonal linear system of size n on a parallel computer with n processors using O ( log n ) parallel arithmetic steps. Here we describe a limited processor version of the recursive doubling algorithm for the solution of tridiagonal linear systems using O ( n / p + log p ) parallel arithmetic steps on a parallel computer with p

Abstract: In this paper, parallel forms of the Gaussian and Jordan Elimination schemes for solving the tridiagonal linear systems which occur frequently in Computational Mathematics are developed and compared with the existing sequential algorithms.

Abstract: Two versions of an algorithm for finding the eigenvalues of symmetric, tridiagonal matrices are described. They are based on the use of the Sturm sequences and the bisection algorithm. The algorithms were implemented on the FPS T-Series. Some speedup factor results are presented.

Abstract: A novel general approach to round-off error analysis using the error complexity concepts is described. This is applied to the analysis of the Gaussian Elimination and Gauss-Jordan scheme for solving linear equations. The results show that the two algorithms are equivalent in terms of our error complexity measures. Thus the inherently parallel Gauss-Jordan scheme can be implemented with confidence if parallel computers are available.

Abstract: Algorithm for solving the difference equations is considered. The difference eq. obtained with approximate factorization for 3-D stable implicit schemes may become unstable or conditionally stable. If proper Jacobian matrix splitting is used stable approximately factored scheme can be obtained. In order to remove the factorization error en efficient nested iterative method is suggested. In this method only tridiagonal matrix inversions are needed. Amount of computation is much reduced at each time step.