A Search for Better Linear Multistep Methods for Stiff Problems

Abstract : For arbitrary k >/- 1 and alpha epsilon (O,pi/2), A(alpha)-stable k-th order k-step formulas exist, so that in an ODE solver, a can be an extra parameter used to identify among a family of methods of order k the A(alpha)-stable method that should be used for the particular problem. Two measures for assessing the accuracy of k-th order k-step formulas are proposed. The problem of finding the upper bound on the angle of absolute stability for the k-th order k-step formulas having the same accuracy (with respect to one of the measures) is considered. Analytical results are obtained for k = 1, 2, 3 whereas a numerical search is used for the cases when k = 4, 5, 6, 7.