ABSTRACT
In the previous Chapter we described discrete approximation of differential equations by using both the Taylor series expansion and the control-volume approach about a grid point. Thus the transport problem governed by a single or a set of differential equations and boundary conditions can be approximated by a system of algebraic equations. If the resulting system is linear and the algebraic equations are not so many, they can readily be solved by using any one of the standard computer subroutines for solving system of algebraic equations. However, if the number of equations to be solved is very large and/or the equations are nonlinear, one needs to examine the nature of the resulting system of equations, because the proper choice of the computer subroutine for solving sets of algebraic equations is strongly affected by the following considerations:
Whether the problem is linear or nonlinear,
Whether the coefficient matrix is tridiagonal, full or sparse (i.e., large percentage of entries are zero),
Whether the number of operations involved in the algorithm is so large as to give rise to excessive accumulation of round-off errors,
Whether the coefficient matrix is “diagonally dominant”,
Whether the coefficient matrix is ill-conditioned (i.e., small changes in the coefficients, such as those introduced by round off errors produce large changes in the solution).
