NUMERICAL SOLUTION OF TWO-POINT BOUNDARY-VALUE PROBLEMS Thesis By Andrew Benjamin White, Jr. In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 91109 1974 (Submitted March 13, 1974) Acknowledgements The author wishes to thank Professor Herbert B, Keller for his patient guidence and invaluable assistance in the preparation of this thesis. His encouragement and enthusiasm were essential to the completion of this work. The opportunity to carry out this research was provided by California Institute of Technology Graduate Teaching Assistantships for which the author is grateful. He also wishes to express his appreciation to the entire Department of Applied Mathematics, both faculty and students, for providing an invigorating climate in which to pursue this work. The author wishes to thank Mrs. Janet Smith and Mrs. Mary Beth Briggs for their expert typing of a detailed and tedious manuscript. ABSTRACT The approximation of two-point boundary-value problenls 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. ~ewton'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. Table of Contents TITLE PAGE Introduction Chapter 1. Linear Two-Point Boundary-Value Problems 1. Existence Theory 2. Numerical Methods 3. Convergence for Linear Boundary- Value Problems Chapter 2. Difference Schemes 1. Taylor Series 2. Quadrature 3. Gap Schemes 4. Other Schemes 5. Stability of Triangular Schemes 6. Asymptotic Error Expansions 7. Increased Accuracy Chapter 3. Parallel Shooting 1. Parallel Shooting 2. Method of Complementary Functions 3. Method of Adjoints Chapter 4. Nonlinear Two-Point Boundary-Value Problems 1. Finite Difference Schemes 2. Existence of Solutions 3. Newton's Method 4. Equivalence of Shooting and Implicit Schemes Chapter 5. Block Tridiagonal Matrices 1. Block LU Decomposition 2. A Split Decomposition Chapter 6. A Numerical Example: Plane Couette Flow 1. Isolated Solution 2. Numerical Procedure 3. Numerical Results Appendix A References Introduction This thesis deals with the application of finite difference schemes to two-point boundary-value problems. The assumption is made throughout that these boundary-value problems have isolated solutions; that is, the homogeneous, linearized problem has only the trivial solution, The general theory developed places no restrictions on the form of the difference equations, In Chapter 1, the application of an arbitrary, consistent difference scheme to a linear boundary-value problem is treated. In the main theorem of this chapter, Theorem 1.16, the stability of the discrete boundary-value problem is shown to be equivalent to the stability of the associated discrete initial-value problem. This associated initial-value problem employs the same difference equations to approximate the differential equation, but initial conditions replace the boundary conditions. From this result, it is clear that a simple shooting method is, in fact, a specific procedure for solving the discrete boundary-value problem. Special emphasis is placed on one-step difference schemes in Chapter 2, High order accurate difference approximations are developed using both Taylor series and the integral form of the differential equation, In particular, one-step schemes of arbitrary order are derived which require the evaluation of a minimum number of new functions (e.g derivatives of A(t), f (t)). The equivalence shown in Chapter 1 is used to examine the stability of triangular difference schemes. In Chapter 3, the equivalence result of Theorem 1.16 is general- ized to include a l l parallel shooting methods. Theorem 3.22 shows that these methods are each a particular procedure for solving the equations derived from approximating linear boundary-value problems. The Method of Complementary Functions is examined in detail as an example of methods for solving problems with separated boundary conditions. The Method of Adjoints is also considered and it is shown that this method is not in general equivalent to the discrete boundary-value problem, Nonlinear boundary-value problems are dealt with in Chapter 4. The difference schemes examined in Chapter 2 are generalized to be applicable to nonlinear differential equations. Following Keller [ 6 1, existence and uniqueness of these discrete approximations is shown. We note that ~ewton'sm ethod converges quadratically. Chapter 5 is concerned with the practical problem of solving the systems of algebraic equations arising from the approximation of boundary-value problems with separated boundary conditions. These equations are written in block tridiagonal form, Mx = b. The special zero structure of this system is exploited to show that, with an appropriate row switching strategy, such a matrix possesses a simple block LU decomposition if and only if M is nonsingular. A numerical example is presented in Chapter 6. The equations considered model plane Couette flow. The Gap4 scheme, as derived in Chapter 4, is used to discretize the nonlinear boundary-value problem and ~ewton'sm ethod is employed to solve the resulting set of nonlinear equations. A consistent effort is made to use o to represent zero or a zero vector and O'for zero matrices, except in tables or equation numbers. The numbering of theorems, equations, or tables is done consecutively throughout each chapter. Chapter I Linear Two-Point Boundary-Value Problems 1. Existence Theory We consider the system of n first-order, linear ordinary differential equations: where u, f, 6 are n-vectors and A,B ,B are n x n matrices. Before 0 1 proceeding to the numerical approximation of (l.la,b), we present an existence and uniqueness result convenient for our purposes. Theorem 1.2. Let A(t) E cm[o,l] for some m 5 o. Define the fundamental matrix ~ ( t a) s the solution of - X' (t) A(t) X(t) = o X(O) = I. Then for each f(t) E cm[o,l] and 6 E E", problem (l.la,b) has a . , unique solution y (t) E cm+l[ 0 l] iff [BO+BIX (1) ] is nonsingular Proof: The solution to the initial-value problem is in general Uniqueness for the initial-value problem insures that X(t) is nonsingular on [o,l]. The boundary-value problem (l.la,b) has a solution if and only if we can define an n-vector r such that y(t) satisfies the boundary condition This requires Thus, (l.la,b) has a unique solution if and only if [B + B X(1) 1 is 0 1 nonsingular. That y(t) E cm+'[o,l] is an observation from the form of differential equation (1.la). 2. Numerical Methods Here we discuss some standard concepts of numerical analysis and develop some notation. In approximating the solution of i=J (l.la,b), wewillemployanetofpoints It,} on [o,l] andanet i=J 1 function {v, defined on this net. Each v is an n-vector and we define V such that i
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