ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING STIFF INITIAL VALUE PROBLEMS

This paper modified an existing 3–point block method for solving stiff initial value problems. The modification leads to the derivation of another 3 – point block method which is suitable for solving stiff initial value problems. The method approximates three solutions values per step and its order is 5. Different sets of formula can be generated from it by varying a parameter ρ ϵ (-1, 1) in the formula. It has been shown that the method is both Zero stable and A–Stable. Some linear and nonlinear stiff problems are solved and the result shows that the method outperformed an existing method and competes with others in terms of accuracy

Improved Two-Point Block Backward Differentiation Formulae for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.

Construction of Stable High Order One-Block Methods Using Multi-Block Triple

This paper deals with the construction of l-stable implicit one-block methods for the solution of stiff initial value problems. The constructions are done using three different multi-block methods. The first multi-block method is composed using Generalized Backward Differentiation Formula (GBDF) and Backward Differentiation Formula (BDF), the second is composed using Reversed Generalized Adams Moulton (RGAM) and Generalized Adams Moulton (GAM) while the third is composed using Reversed Adams Moulton (RAM) and Adams Moulton (AM). Shift operator is then applied to the combination of the three multi-block methods in such a manner that the resultant block is a one-block method and self-starting. These one-block methods are up to order six and with at order ten. Numerical experiments show that they are good for solving stiff initial problems.

A Family of High Order One-Block Methods for the Solution of Stiff Initial Value Problems

In this paper, we construct a family of high order self-starting one-block numerical methods for the solution of stiff initial value problems (IVP) in ordinary differential equations (ODE). The Reversed Adams Moulton (RAM) methods, Generalized Backward Differentiation Formulas (GBDF) and Backward Differentiation Formulas (BDF) are used in the constructions. The E-transformation is applied to the triples and a family of self-starting methods are obtained. The family is for . The numerical implementation of the methods on some stiff initial value problems are reported to show the effectiveness of the methods. The computational rate of convergence tends to the theoretical order as h tends to zero.

Time parallelization scheme with an adaptive time step size for solving stiff initial value problems

In this paper, we introduce a practical strategy to select an adaptive time step size suitable for the parareal algorithm designed to parallelize a numerical scheme for solving stiff initial value problems. For the adaptive time step size, a technique to detect stiffness of a given system is first considered since the step size will be chosen according to the extent of stiffness. Finally, the stiffness detection technique is applied to an initial prediction step of the parareal algorithm, and select an adaptive step size to each time interval according to the stiffness. Several numerical experiments demonstrate the efficiency of the proposed method.