Numerical Analysis of Ordinary Differential Equations and Its Applications

The following sections are included:

• Preface

• CONTENTS

It is well known that eight-stage explicit Runge–Kutta formulas are of order at most six. However, by taking the limit as the first abscissa approaches zero, the formulas can achieve seventh order. Such formulas are called limiting formulas, which requre the evaluations of the second derivatives of the solution. In this paper, eight-stage seventh order limiting formulas using the second derivatives are derived. And based on these limiting formulas, new eight-stage numerically seventh order methods without derivatives are proposed.

Collocation Runge-Kutta formulae, a dominant class of implicit methods, are considered for numerical initial value problems of ODEs. Since they are fully characterized by their abscissae, we propose a systematical way to generate collocation Runge-Kutta formulae of the same number of stages by a gradual change of abscissae. They increase their orders as the changing, finally to coincide with the Butcher-Kuntzmann formula of the specified number of stages. Their A-stability is also investigated to represent the stability factor with the abscissae.

A certain type of P-stable block method is derived for solving second order initial value problem y″ = f(x, y). The block method considered here computes the numerical solutions simultaneously at the two points of x, and is easily parallelisable. Some technique to reduce the local truncation error of the method is also developed.

In this paper a special class of Hermite-Birkhoff quadratures is investigated and applied to numerical method for solving directly any higher order ordinary differential equation. That is, if an incidence matrix E = (e_ij) defined by e_ij = 0 or 1, (1≤i≤2, 0≤j≤n - 1), and is poised in the viewpoint of the lacunary interpolation theory, it is shown that there exists a quadrature formula in the form

We consider the linear system Ax = b derived from singular perturbation problems. To solve such non-symmetric linear problems, we propose a generalized SOR method, which we have named the "improved SOR method with orderings". We use three ideas, that is, orderings, variable relaxation parameters and a not necessarily strictly upper triangular splitting matrix U in A = D - L - U. The basic theorem, the selection of the relaxation parameters, orderings and several numerical experiments are also presented.

The behavior of the difference between the values of the predictor and of the corrector for the Adams predictor–corrector method in the P(EC)^m mode is analysed. This leads not only to an accurate estimation of local truncation errors but also to that of global truncation errors.

A new algorithm is proposed to solve differential-algebraic equations. The algorithm is an extension of the algorithm of general purpose HIDM (higher order implicit difference method). A computer program named HDMTDV and based on the new algorithm is constructed and its high performance is proved numerically through several numerical computations, including index-2 problem of differential-algebraic equations and connected rigid pendulum equations.

The new algorithm is also secular error free when applied to dissipationless dynamical systems. This nature is demonstrated numerically by computation of the Kepler motion. The new code can solve the initial value problem

A-stable semi-explicit methods are constructed for differential-algebraic systems of index 1 and for those of index 2 and their convergence is shown.

One of the outstanding problems in the numerical simulation of mechanical systems is the development of efficient methods for dealing with highly oscillatory systems. These types of systems arise for example in vehicle simulation in modelling the suspension system or tires, in some models for contact and impact, in flexible body simulation from vibrations in the structural model, and in molecular dynamics. Simulations involving high frequency vibration can take a huge number of time steps, often as a consequence of oscillations which are not physically important. On the other hand, the components causing the oscillations cannot usually be eliminated from the model because in some situations they are critical to the simulation. The equations of motion of a multibody mechanical system are described by a system of differential-algebraic equations (DAEs). In this paper, we will explore two types of methods. The first class of methods damps out the oscillation via highly stable implicit methods. Even in this relatively simple approach, unforseen problems may arise for Newton iteration convergence, due to the nonlinearities. The second class of methods involves formulating the multibody system in such a way that the oscillations are determined by a linear subsystem, which can potentially be solved rapidly via a number of different methods.

This paper is concerned with the existence and the uniqueness of quasiperiodic solutions to quasiperiodic nonlinear differential equations in the neighborhood of quasiperiodic solution to linear differential equation or in the neighborhood of Galerkin approximation to the nonlinear differential equations. By defining the generalized exponential dichotomy, our Theorem 4 will be useful independently whether the nonlinearity is weak or not.

Some numerical results are shown. These results show that our analysis is useful for mathematical investigation of quasiperiodic phenomena such as the design of communication circuits.

A natural Runge-Kutta (RK) method is a RK method which has a special continuous extesion. Any one-step collocation method is equivalent to one of such methods. In this paper, we consider the application of natural RK methods to delay differential equations (DDEs) which have a constant delay, and discuss their numerical stability applied to several types of test equations whose zero solution is stable for arbitrary value of the delay. As a result, we show that an A-stable method preserves the asymptotic property of the analytical solution of a DDE coupled with a difference equation (i.e., a delay-differential-algebraic equation).

A method of computer assisted existence proof is discussed for solutions of nonlinear boundary value problems. In 1966, Urabe has presented a convergence theorem for a certain simplified Newton method. Urabe's theorem is essentially based on Banach's contraction mapping theorem. In this paper, reformulation of Urabe's theory using the interval analysis is presented. It is shown that a sharp error estimation can be obtained by this reformulation.

The development of software technology of automatic differentiation and interval calculus has enabled us to numerically construct an interval solution for the initial-value problem of ordinary differential equations. In this paper, we will report how we applied this method to several problems as a practical technique and what observations we got.

In this paper a numerical validation method for normal form simultaneous first order differential equations is discussed. Based on Lohner's method and interval functoid, a new inclusion algorithm for initial value problem is given. For the algorithm, two types of arithmetics of power series is defined.

Simulation for some stochastic integral processes is required in numerical solutions for stochastic differential equations (SDEs). The stochastic part of the error in simulation is considered, especially for the Wiener process W(t) and the Wiener integral process ∫ sdW(s) as the basics. Several weak numerical schemes are applied for a good approximation of statistical quantities of the solutions. The results show that the error depends on the number of trajectories, not on the stepsize. The way to realize basic integral processes is discussed.