In this study, we develop a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides. The method is a generalization of the parallel dichotomy algorithm for solving systems of linear equations with tridiagonal matrices \cite{terekhov:Dichotomy}. Using this approach, we propose a parallel realization of the domain decomposition method (\mbox{the Schur} complement method). The calculation of acoustic wave fields using the spectral-difference technique improves the efficiency of the parallel algorithms. A near-linear dependence of the speedup with the number of processors is attained using both several and several thousands of processors. This study is innovative because the parallel algorithm developed for solving block-tridiagonal systems of equations is an effective and simple set of procedures for solving engineering tasks on a supercomputer.

pdf/a-fast-parallel-algorithm-for-solving-block-tridiagonal-3eiqedtqde.pdf

A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method

In this study, we develop a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides. The method is a generalization of the parallel dichotomy algorithm for solving systems of linear equations with tridiagonal matrices [1] . Using this approach, we propose a parallel realization of the domain decomposition method (the Schur complement method). The calculation of acoustic wave fields using the spectral-difference technique improves the efficiency of the parallel algorithms. A near-linear dependence of the speedup with the number of processors is attained using both several and several thousands of processors. This study is innovative because the parallel algorithm developed for solving block-tridiagonal systems of equations is an effective and simple set of procedures for solving engineering tasks on a supercomputer.

The popular sequential Thomas algorithm for the numerical solution of tridiagonal linear algebraic equation systems is extended on a class of quasi-block-tridiagonal equation systems arising from finite-difference discretisations of boundary value and initial boundary value problems of reaction-migration-advection-diffusion type in one space dimension, occurring in electrochemistry. The extension allows for a simultaneous consideration of: (a) multiple space intervals with common boundaries; (b) additional algebraic or differential-algebraic equations coupled with mixed boundary conditions, that may express e.g. adsorption at the boundaries; (c) three-point finite-difference approximations to the gradients of the solutions of the initial/boundary value problems at the boundaries; (d) periodic or non-periodic boundary conditions at the external boundaries. The resulting equation matrix may include nonzero off-diagonal corner blocks associated with periodic boundary conditions, may be locally block-pentadiagonal at a number of isolated rows corresponding to internal spatial boundaries, and its blocks may have variable dimensions. Testing calculations are performed.

Extension of the Thomas algorithm to a class of algebraic linear equation systems involving quasi-block-tridiagonal matrices with isolated block-pentadiagonal rows, assuming variable block dimensions

Essentially, only two implicit methods will be described here, but with extensions that make them more useful. They are derived from the implicit methods described for odes in Chap. 4, BI and the trapezium method. These have different names in the pde context, as will be seen.

The Commonly Used Implicit Methods

Brute force digital simulation

Recurrence solution of a block tridiagonal matrix equation with Neumann, Dirichlet, mixed or periodic boundary conditions

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Recurrence solution of a block tridiagonal matrix equation with Neumann, Dirichlet, mixed or periodic boundary conditions