Functionally-fitted Block Methods For Ordinary Differential Equations

Ordinary differential equations(ODEs) are widely used in producing models in physical sciences, biological sciences, engineering and economics. Many ordinary differential equations are too complex for us to seek for an analytic solution. For practical purposes, however,numerical approximations to the solutions are often sufficient. Numerical methods are methods used to find numerical approximations to the solutions of ODEs.Numerous numerical methods including Runge-Kutta methods, multistep methods and block methods have been developed for solving the initial value problems of ordinary differential equations. These methods have constant coefficients and can integrate exactly a set of algebraic polynomials up to a certain degree.It is possible to design variable coefficient methods that are also exact with functions other than algebraic polynomials. Examples include trigonometric polynomials, exponential functions, as well as mixed algebraic and trigonometric polynomials. Such variable coefficient methods are proved to be more advantageous than the polynomial based methods for the special situations where some knowledge of the problems is known in advance especially when the solutions exhibit a prominent periodic or oscillatory behaviour.This thesis attempts to construct a family of functionally-fitted block(FFB) methods which generalizes the class of block implicit one-step methods and discuss the convergence and the stability of the methods for initial value problems for the first and second order differential equations. It is organized as follows.In Chapter 1, we present a brief introduction to basic concepts and theories of ordinary differential equations. Chapter 2 is devoted to a summary of classical numerical methods(including Runge-Kutta methods and linear multistep methods), block methods and functionally-fitted methods for solving initial value problems in ODEs.Chapter 3 establishes the basic theory of the functionally-fitted block methods for first order ordinary differential equations. We briefly recall the block implicit one-step methods first proposed by Shampine and Watts [101] and describe the new class of FFB methods whose coefficient matrix depends on the integration time and the step size. We derive two different sufficient conditions to ensure the existence of the FFB methods, discuss their equivalence to collocation methods in a special case, and study their independence on integration time for a set of separable basis functions. Some basic characteristics of the FFB methods are presented by means of Taylor series expansions and the order of accuracy is obtained. Finally, we present some numerical results to demonstrate the validity of the functionally-fitted block methods.In Chapter 4, we propose a new type of functionally-fitted block methods for solving the second order ordinary differential equations. The basic theory for the FFB methods is established. After describing the new class of FFB methods whose coefficients depend on the integration time and the step size, we derive a sufficient condition to ensure the existence of the FFB methods, and study their independence on integration time for a set of separable basis functions. We present some basic characteristics of the FFB methods by means of Taylor series expansions and obtain the order of accuracy. Some numerical experiments are conducted to compare the FFB methods with a constant coefficient block hybrid method and other variable coefficient methods in the literature, and show that the FFB methods are competitive and efficient.Some conclusions and remarks on future work are included in Chapter 5.