Numerical Solution of Two-Point Boundary Value Problems

Citation

White, Andrew Benjamin, Jr. (1974) Numerical Solution of Two-Point Boundary Value Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R9VN-9C49. https://resolver.caltech.edu/CaltechETD:etd-01312007-163410

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.

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