Block backward differentiation alpha-formulas for solving stiff ordinary differential equations

A new family of block methods, namely block backward differentiation alpha-formulas (BBDF-) are developed for solving first and second order stiff ordinary differential equations (ODEs) directly. By selecting the appropriate values of parameter that can be controlled by user, the derived methods give better approximation compared to the existing methods. Initially, the derivation of BBDF- using constant and variable step size approach for solving first order stiff ODEs is presented. The consistency and zero stability that lead to the convergence properties are discussed theoretically. Meanwhile, the stability regions are displayed to show that the derived methods are A-stable for certain values of. Numerical results reveal the superiority of the derived formulas in terms of total number of steps, accuracy and computation time. Subsequently, the BBDF- is constructed for solving second order stiff ODEs directly. This method is specially designed to cater the second order ODEs without reducing it into the first order. The convergence aspects are investigated and the stability region is illustrated to verify the suitability of the method in solving stiff problems. Numerical results demonstrate the advantage of the method in terms of execution time due to its capability as direct solver. Furthermore, the BBDF- is formulated using variable step size scheme for solving second order stiff ODEs directly. In order to describe the whole process of implementation, the numerical algorithm is exhibited. The results indicate that the developed method has advantage in terms of accuracy and total number of step. Finally, the application of derived methods in damped oscillation problems is presented. To test the performance of the methods, several experiments on over-damped, critically-damped and under-damped oscillation in mass-spring systems are conducted. In conclusion, the derived methods can be used as viable alternative solver for stiff ODEs and real-life problem.

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