A class of two stage multistep methods in solutions of time dependent parabolic PDEs

Abstract

In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, \((p=4,5,6,7)\) and \(A(\alpha \))-stable for order p, \((p=8,9,\ldots , 14).\) Compared to the existing BDF based method, i.e. class \(2+1,\) hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF.