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

G Protein-Coupled Receptor-Mediated Calcium Signaling in Astrocytes

“…The two files astrocyte_models.h and astrocyte_models.cpp contains the core G-ChI model implementation in C/C++11, while the class Astrocyte in astrocyte_models.py provides the Python interface to simulate the G-ChI model. The model was integrated by a variable-coefficient linear multistep Adams method in Nordsieck form which proved robust to correctly solve stiff problems rising from different parameter choices (Skeel, 1986). Model fitting is provided by gchi_fit.py and relies on the PyGMO 2.6 optimization package (https://github.com/esa/pagmo2.git).…”

A new stepsize change technique for Adams methods

“…3 The local truncation error Although the matrix form (13)- (14) of the AMk method will be useful to study the stability of the methods, for practical purposes it is more convenient to write the equations in the equivalent form…”

“…, k, we have the AMk method with the interpolation technique for stepsize changing. Other techniques have been proposed by Skeel [14], Gupta and Wallace [8] and Jackson and Sack-Davis [10].…”

Adaptive time-stepping and computational stability

“…The former argument never holds except possibly for ODEs with only a very few equations; the latter is not very significant either, as a small stepsize variation can easily be accommodated in the Newton iterative process, see [6] or the actual strategy used in Dassl [2], which is based on an over/under-relaxation, compensating for the changed stepsize. The only more significant argument for a cut-out seems to be that there are some multistep method implementations whose coefficients have singularities for certain small stepsize ratios, [1,9]. This is however not the case for BDFs.…”

“…Moreover, by introducing a smooth nonlinear transformation, we can employ a smooth limiter that eliminates the previous discontinuities. First ρ n is computed from (9). Then this value is limited so that it is bounded away from 0 and ∞ by a transformationρ n = ω(ρ n ).…”