Additional degrees of parallelism within the Adomian decomposition method

Abstract

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.