Abstract
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.
Highlights
IntroductionPhysical and social sciences are reduced to quantifiable form through mathematical modelling involving systems of ordinary differential equations (ODEs)
Many problems in engineering, physical and social sciences are reduced to quantifiable form through mathematical modelling involving systems of ordinary differential equations (ODEs).These problems sometimes exhibit a phenomenon known as stiffness
By solving | R( H )| < 1, we found that the singly diagonally implicit block backward differentiation formulas (SDIBBDF) method is stable everywhere except when
Summary
Physical and social sciences are reduced to quantifiable form through mathematical modelling involving systems of ordinary differential equations (ODEs). Conventional RK methods evolutionized to form the best solver for ODEs. The commonly improved RK methods involved an implementation of singly diagonally implicit properties which had proved to produce better computational time when compared with the existing methods. Only one evaluation of the Jacobian and one LU decomposition will be needed for each time step By having these properties, degree of implicitness can be reduced as it involved less computational process which will results in less execution time [21]. Developing the SDIBBDF method involves the hybrid process of implementing qualities from SDIRK method to block multistep method As we know, both methods are of different families our main concern is the compatibility of the derived method to solve for stiff ODEs. Derivation of the SDIBBDF method is shown
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