Block one-step methods for solving stiff differential equations - Mohd Zabidi, Muhammad Izzat Zakwan (2014)
In this research, both stiff ordinary differential equations (ODEs) and parabolic partial differential equation (PDEs) are solved using the A-stable one-step block method with Newton’s iteration with constant step size. Two-point block one-step method and three-point block one-step method had been proposed in this research. These two methods are used to approximate the solutions for stiff ODEs and parabolic PDEs at two and three points simultaneously. The implementation of these methods will be in predictor and corrector mode. The predictor formulae is formulated from the modified block method itself. Newton’s iteration is adapted in implementation of the block methods. The order, error constant, convergence and stability of each method are also discussed. This study also focused on solving parabolic PDEs. In order to solve parabolic PDEs using the proposed methods, we reduced the form of parabolic PDEs into ODEs by discretizing the parabolic equation using method of line. To illustrate the applicability of the proposed method, several numerical results are shown and compared with the results obtained by the existing methods In conclusion, the proposed methods are suitable for solving stiff ordinary differential equations at varies stepsizes especially when the stepsizes are larger. Other than that, the proposed method also appropriate for solving stiff parabolic partial differential equations due to acceptable results that had been produced.