Generalized Multistep Predictor-Corrector Methods
Journal of the ACM (JACM)
An exponential method of numerical integration of ordinary differential equations
Communications of the ACM
Numerical Methods for Scientists and Engineers
Numerical Methods for Scientists and Engineers
Cyclic composite multistep predictor-corrector methods
ACM '69 Proceedings of the 1969 24th national conference
A review of recent developments in solving ODEs
ACM Computing Surveys (CSUR) - Annals of discrete mathematics, 24
A stiffly stable integration process using cyclic composite methods
ACM Transactions on Mathematical Software (TOMS)
| Hi-index | 0.00 |
Consider the set of multistep formulas ∑l-1jmn-k &agr;ijxmn+j - h ∑l-1jmn-k&bgr;ijxmn+j = 0, i = 1, ···, l, where xmn+j = ymn+j for j= -k, ···, -1 and xn = ƒn = ƒ(xn , tn). These formulas are solved simultaneously for the xmn+j with j = 0, ···, l - 1 in terms of the xmn+j with j = -k, ··· , - 1, which are assumed to be known. Then ymn+j is defined to be xmn+j for j = 0, ··· , m - 1. For j = m, ··· , l - 1, xmn+j is discarded. The set of y's generated in this manner for successive values of n provide an approximate solution of the initial value problem: y = ƒ(y, t), y(t0) = y0. It is conjectured that if the method, which is referred to as the composite multistep method, is A-stable, then its maximum order is 2l. In addition to noting that the conjecture conforms to Dahlquist's bound of 2 for l = 1, the conjecture is verified for k = 1. A third-order A-stable method with m = l = 2 is given as an example, and numerical results established in applying a fourth-order A-stable method with m = 1 and l = 2 are described. A-stable methods with m = l offer the promise of high order and a minimum of function evaluations—evaluation of ƒ(y, t) at solution points. Furthermore, the prospect that such methods might exist with k = 1—only one past point—means that step-size control can be easily implemented