Implicit Block Backward Differentiation Formulas Research Articles

A New Fixed Coefficient Diagonally Implicit Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Stiff initial value problems in ordinary differential equations occur when solution components evolve at varying rates, posing challenges for traditional computational methods. Specialized techniques are crucial for maintaining accuracy and stability during rapid transitions, emphasizing their significance in developing reliable numerical algorithms across scientific and engineering applications. This study aims to develop a new fixed coefficient 3-point diagonally implicit block backward differentiation formula for the numerical solution of first order stiff initial value problems. The method is constructed by integrating a triangular matrix into the coefficient matrix of an existing extended 3-point super class of block BDF for solving stiff initial value problems. The selection of a fixed coefficient within the interval accompanies this integration to ensure optimal stability. The method is found to order five. Stability analysis indicates that the method is consistent, zero-stable, and almost A-stable, validating its applicability to stiff initial value problems. Implementation of the method involves Newton’s iteration, and a code in the C programming language is devised to demonstrate its effectiveness. Comparative examination of numerical outcomes with the existing 3BBDF and 3ESBBDF methods highlights the proposed method's enhanced accuracy and reduced computation time.

Convergence and Order of the 2-Point Diagonally Implicit Block Backward Differentiation Formula with Two Off-Step Points

The development and formulation of a most reliable and efficient numerical schemes for the integration of stiff systems of ordinary differential equations in terms of order, convergence, stability requirements, accuracy, and computational expense has been a major challenged in the study of modern numerical analysis. In this paper, the order and convergence properties of the 2-point diagonally implicit block backward differentiation formula with two off-step points for solving first order stiff initial value problems have been studied, the method was derived and found to be of order five. The necessary and sufficient conditions for the convergence of the method have also been established. It has shown that the 2-point diagonally implicit block backward differentiation formula with two off-step points is both consistent and zero stable, having satisfied these two conditions of consistency and that of zero stability, it is therefore concluded that the method converges and suitable for the numerical integration of stiff systems.

On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six

In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.

Stability Analysis of Singly Diagonally Implicit Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

In this research, a singly diagonally implicit block backward differentiation formulas (SDIBBDF) for solving stiff ordinary differential equations (ODEs) is proposed. The formula reduced a fully implicit method to lower triangular matrix with equal diagonal elements which will results in only one evaluation of the Jacobian and one LU decomposition for each time step. For the SDIBBDF method to have practical significance in solving stiff problems, its stability region must at least cover almost the whole of the negative half plane. Step size restriction of the proposed method have to be considered in order to ensure stability of the method in computing numerical results. Efficiency of the SDIBBDF method in solving stiff ODEs is justified when it managed to outperform the existing methods for both accuracy and computational time.

Corrigendum to “Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems”

Corrigendum to “Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems”

Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

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 methods.