Numerical solution of two-point boundary-value problems

Abstract

The approximation of two-point boundary-value problems by general finite difference schemes is treated. A necessary and sufficient condition for the stability of the linear discrete boundary-value problem is derived in terms of the associated discrete initial-value problem. Parallel shooting methods are shown to be equivalent to the discrete boundary-value problem. One-step difference schemes are considered in detail and a class of computationally efficient schemes of arbitrarily high order of accuracy is exhibited. Sufficient conditions are found to insure the convergence of discrete finite difference approximations to nonlinear boundary-value problems with isolated solutions. Newton's method is considered as a procedure for solving the resulting nonlinear algebraic equations. A new, efficient factorization scheme for block tridiagonal matrices is derived. The theory developed is applied to the numerical solution of plane Couette flow.