Comparing numerical methods for the solutions of systems of ordinary differential equations

AbstractIn this article, we implement a relatively new numerical technique, the Adomian decomposition method, for solving linear and nonlinear systems of ordinary differential equations. The method in applied mathematics can be an effective procedure to obtain analytic and approximate solutions for different types of operator equations. In this scheme, the solution takes the form of a convergent power series with easily computable components. This paper will present a numerical comparison between the Adomian decomposition and a conventional method such as the fourth-order Runge-Kutta method for solving systems of ordinary differential equations. The numerical results demonstrate that the new method is quite accurate and readily implemented

Additional degrees of parallelism within the Adomian decomposition method

4th International Conference on Computational Engineering (ICCE 2017), 28-29 September 2017, DarmstadtThis is the author accepted manuscript. The final version is available from Springer via the DOI in this record.The trend of future massively parallel computer architectures challenges the exploration of additional degrees of parallelism also in the time dimension when solving continuum mechanical partial differential equations. The Adomian decomposition method (ADM) is investigated to this respects in the present work. This is accomplished by comparison with the Runge-Kutta (RK) time integration and put in the context of the viscous Burgers equation. Our studies show that both methods have similar restrictions regarding their maximal time step size. Increasing the order of the schemes leads to larger errors for the ADM compared to RK. However, we also discuss a parallelization within the ADM, reducing its runtime complexity from O(n^2) to O(n). This indicates the possibility to make it a viable competitor to RK, as fewer function evaluations have to be done in serial, if a high order method is desired. Additionally, creating ADM schemes of high-order is less complex as it is with RK.The work of Andreas Schmitt is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universit¨at Darmstadt