Hamisu Musa

A new variable step size block backward differentiation formula for solving stiff initial value problems

A new block backward differentiation formula of order 4 with variable step size is formulated. By varying a parameter in the formula, different sets of formulae with A-stability property can be generated. At the cost of an additional function evaluation, the accuracy of the method is seen to outperform some existing backward differentiation formula algorithms. The strategy involved in controlling the step size ratio is also described. The problems tested with the method show its efficiency in solving stiff initial value problems.

An improved 2-point block backward differentiation formula for solving stiff initial value problems

A new block method that generates two values simultaneously is developed for the integration of stiff initial value problems. The method is proven to be A – stable and is a super class of the 2 – point block backward differentiation formula (BBDF). A comparison is made between the method, 1 point backward differentiation formula (BDF) and the 2 point BBDF methods. The numerical results indicate that the new method outperformed the 1 point BDF and the 2 point BBDF methods in terms of accuracy and stability. The total number of steps to complete the integration by the 1 point BDF method is reduced to half. Computation time for the method is also competitive.

An Accurate Block Solver for Stiff Initial Value Problems

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.