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 ...

On the stability and error structure of BDF schemes applied to sectorial problems – p. 1/17

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas DI2BBDF for solving first-order initial value problem IVP with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing ...

We determine new, more favourable, and in a sense optimal, multipliers for the threeand five-step backward difference formula (BDF) methods. We apply the new multipliers to establish stability of these methods as well as of their implicit–explicit counterparts for parabolic equations by energy techniques, under milder conditions than the ones recently imposed in [4, 1].

This is a survey of essentially normal operators and related developments. There is an overview of Weyl–von Neumann theorems about expressing normal operators as diagonal plus compact operators. Then we consider the Brown–Douglas–Fillmore theorem classifying essentially normal operators. Finally we discuss almost commuting matrices, and how they were used to obtain two other proofs of the BDF t...

There are three important steps in the proof the BDF theorem for essentially normal operators having essential spectrum X ⊆ C. (1) Ext(X) has a neutral element. (2) Ext(X) is a group (i.e., has inverses). (3) Ext(X) depends only on the homotopy class of X. In this lecture we will exhibit the neutral element of Ext(X) and describe the generalization of that result to noncommutative C∗-algebras (...

The Generalized Backward Differentiation methods for solving stiff higher-order ordinary differential equations are described. The convergence, zero stability and consistency of these methods are defined. Next, the zero stability and consistency conditions necessary for convergence are proven. The order for which the methods are zero stable is also determined.