Accuracy of multivalue methods during change of time-step size: Adams-Moulton

Abstract

Multivalue methods are a class of time-stepping procedures for numerical solution of differential equations that progress to a new time level using the approximate solution for the function of interest and its derivatives at a single time level. The methods differ from multistep procedures, which make use of solutions to the differential equation at multiple time levels to advance to the new time level. Multistep methods are difficult to employ when a change in time-step is desired, because the standard formulas (e.g., Adams-Moulton or Gear) must be modified to accommodate the change. Multivalue methods seem to possess the desirable feature that the time-step may be changed arbitrarily as one proceeds, since the solution proceeds from a single time level. However, in practice, changes in the time-step introduce lower order errors or alter the coefficient in the truncation error term. Here, the multivalue Adams-Moulton method is presented based on a general interpolation procedure. Modifications required to retain the high-order accuracy of these methods during a change in time-step are developed. Additionally, a formula for the unknown initial derivatives is presented. Finally, two examples are provided to illustrate the potential merit of the modification to the standard multivalue methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partials Differential Eq 16: 312–326, 2000