Comparing the Level of Stability between Standard (Single-Step) and Multi-Step Methods in Numerical Ordinary Differential Equations
Adetunji, A. B. & Adeleke, I.A.
Department of Computer Science and Engineering,
Lautech, Ogbomoso and Computer Science Department, EACOED, Oyo
abadetunji@lautech.edu.ng and adeleke_israel@yahoo.com
ABSTRACT
In solving ordinary differential equation (ODE) numerically, there are series of techniques to be employed but the choice of suitable technique to determine their level of stability is a great problem. This research evaluated Runge-Kutta of fourth order as standard method and Adams-Moulton of fourth order as multistep method with two sample problems of ordinary differential equations to compare their performances. The two problems were solved with the two numerical algorithms selected and implemented using C++ programming language as ten iterations on each sample problem were carried out to determine their stability. The results obtained were presented in form of tables which ranged between -1.000000 to 1.9524898 for Adams method while Runges values ranged between -1.000000 to -0.3639740 and their corresponding values ranged between 4.0000000 to 5.7797685 and 4.0000000 to 5.8775982 respectively. The error differences in relation to the exact-solutions were calculated such that Adam’s method ranged between 5.7E-07 to 2.56E-06 while that of Runge’s method ranged between 5.7E-07 to 2.3165 their corresponding error differences ranged between -6.12E-03 to -5.3E-06 and -6.12E-03 to -9.7835E-02 respectively. Graphs were used to present the error differences to clearly see the level of stability of the two numerical algorithms under consideration. The result showed that Adams-Moulton technique is more stable if not absolutely stable than its counterpart.
Keyword: ODE, stability, multistep, standard-method and exact-solution
