Variable stepsize variable order multistep methods for stiff ordinary differential equations
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Backward differentiation methods are used extensively for integration of stiff systems of ordinary differential equations. During the integration, the steplength and order are controlled so that the estimated local error is less than some user prescribed tolerance. There are two techniques commonly used to implement variable stepsize multistep methods. One technique is based on fixed coefficient formulas and the other is based on variable coefficient formulas. The latter does not require past values to be equally spaced and the coefficients are computed during the integration;In this thesis, a class of multistep formulas which includes the backward differentiation formulas as a subclass is considered. These formulas have two first derivative terms compared to one first derivative term in backward differentiation formulas. For these methods, the variable coefficient implementation is used. When the stepsize is fixed, these formulas are stable up to order seven while the backward differentiation formulas are stable up to order six. Some bounds of the stepsize ratios are obtained for the stability of the order two and three methods when the stepsize is allowed to change during the integration. Some numerical bounds of the stepsize ratios for order four and five cases are also given. Selection of the formulas in the above class is done and comparisons are made with the variable coefficient backward differentiation formulas. A computer code which uses the above type formulas of orders one through five is given. Numerical testing and comparison with two other computer codes are made on a set of test problems.