Construction of Variable-Stepsize Multistep Formulas

Abstract

Abstract. A systematic way of extending a general fixed-stepsize multistep formula to a minimum storage variable-stepsize formula has been discovered that encompasses fixed-coefficient (interpolatory), variable-coefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the " interpolatory" stepsize changing technique of Nordsieck leads to a truly variable-stepsize multistep formula (which has implications for local error estimation and formula changing), and it is shown that the " variable-step" stepsize changing technique applicable to the Adams and backward-differentiation formulas has a reasonable generalization to the general multistep formula. In fact, it is shown how to construct a variable-order family of variable-coefficient formulas. Finally, it is observed that the first Dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration.1. Introduction. Multistep methods have been the most successful numerical methods for solving initial-value problems in ordinary differential equations. The selection of a particular formula is often based on a theoretical analysis of fixedstepsize formulas, and yet implementation normally requires the use of a variablestepsize formula. The question of how to extend a formula to variable stepsize is the primary topic of this paper. Existing techniques for varying stepsize are studied, revealing interesting relationships and useful generalizations. At the same time, the results in Skeel [19] on the equivalence between multivalue methods and multistep methods are extended to variable stepsize. There are techniques for varying the stepsize other than the use of variable-stepsize formulas, and these are included in

Thermo-poro-mechanical analysis of landslides: from creeping behaviour to catastrophic failure

“…Here the trapezoidal rule is used, which is of order O(Δt 2 ). In principle it is also possible to use a higher order, more accurate multistep method; however, adaptive time stepping will only be possible at the cost of significantly increased complexity, for example as shown by Skeel (1986).…”

Thermo-poro-mechanical analysis of landslides: from creeping behaviour to catastrophic failure

“…In principle it is also possible to use a higher order, 547 more accurate multistep method; however adaptive time stepping will only be possible at the 548 cost of significantly increased complexity, e.g. Skeel (1986). 549…”