in this paper, we consider the construction of a new class ofnumerical methods based on the backward differentiation formulas(bdfs) that be equipped by including two off--step points. werepresent these methods from general linear methods (glms) pointof view which provides an easy process to improve their stabilityproperties and implementation in a variable stepsize mode. thesesuperiorities are ...

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge–Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this fo...

In this paper, we consider the construction of a new class of numerical methods based on the backward differentiation formulas (BDFs) that be equipped by including two off--step points. We represent these methods from general linear methods (GLMs) point of view which provides an easy process to improve their stability properties and implementation in a variable stepsize mode. These superioritie...

Previous studies of the stability of the second-order backward differentiation formula have concluded that stability is possible only if restrictions are placed on the stepsize ratios, for example, limiting the ratio to some value less than 1 + fi. However, actual implementations of the BDFs differ from the usual theoretical models of such methods; in particular, practical codes use scaled deri...

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...

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ(A) = A−1(eA − I) on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ(A)v can be approximated by Krylov su...