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

Abstract

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.