Вычислительная математика - ИСЭМ СО РАН

COMPUTATION OF STABILITY REGION OF SOME BLOCK METHODS AND
THEIR APPLICATION TO NUMERICAL SOLUTIONS OF ODES
Ming-Gong Lee, Rei-Wei Song, Yu-Hsang Hong
Department of Applied Mathematics, Chung-Hua University, Taiwan
email: mglee@chu.edu.tw
Abstract. Classes of multistage and multistep integration methods which obtain of r new values
at each step are studied. Their stability regions were sketched by MATLAB, and their regions are
almostA(α)-stable. Their applications to numerical solutions of nonstiff and stiff equations were
studied.
Key words: Multistage and multistep methods, A(α)-stable, stiff equations.
1. Introduction
Numerical solutions for ordinary differential equations (ODEs) have great importance in scientific
computation, as they were widely used to model in representing the world. The common methods
used to solve these ODEs are categorized as one-step (multistage) methods and multistep (one stage)
methods, which Runge-Kutta methods represent the former group, and Adams-Bashforth-Molton
method represents the later group. Some multistage are also available in the community. Implicit
one-step method has been studied by Stoller and Morrison [5], Butcher [2], and Shampine and
Watts [6]. We will consider a class of implicit multistep and multistage methods for solving ordinary
differential equations; we can call it block method. It can obtain a block of new values simultaneously
which makes this implicit method more competitive. Our aim is to analyze the stability of this Block
methods though graphing capability of MATLAB. The interval of stability of a numerical method
is especially important in the choice of a method suitable for stiff system. Indeed, for stiff systems,
an interval of stability as large as possible is required to avoid a very restricted stepsize h during
numerical integration. A large interval of stability may be sufficient in the integration of some stiff
ODEs; for example, when the Jacobian matrix of right-hand side function has eigenvalues which are
located in a large, narrow strip along the negative axis. Such equations often arise when a second order
hyperbolic differential equations in semi-discretized with respect to its space variable [7]. In section 2,
we will introduce some known block implicit one-step method used by Shampine, and their numerical
implementation by predictor and corrector method. In section 3, we will show the derivation of some
of the block multistage/multistep method, and their stability regions will be given at section 4. In
section 5, we will give numerical results of these block-type methods by solving some nonstiff and stiff
ODEs, and finally section 6 is the conclusion.
2. Block Implicit One-Step Method
We wish to approximate the solution of a differential equation,
y ′ (x) = f(x, y(x)), y(a) = y a (1)
In the interval of [a, b], suppose functionf is continuous and satisfies
|f(x, y) − f(x, u)| ≤ L |y − z| (2)
on [a, b] × (−∞, ∞) guarantees the existence of a unique solutiony(x) ∈ C 1 [a, b].
208