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

Computational aspects of harmonic wavelet Galerkin methods and an application to a precipitation front propagation model

This article is dedicated to harmonic wavelet Galerkin methods for the solution of partial differential equations. Several variants of the method are proposed and analyzed, using the Burgers equation as a test model. The computational complexity can be reduced when the localization properties of the wavelets and restricted interactions between different scales are exploited. The resulting variants of the method have computational complexities ranging from O(N(3)) to O(N) (N being the space dimension) per time step. A pseudo-spectral wavelet scheme is also described and compared to the methods based on connection coefficients. The harmonic wavelet Galerkin scheme is applied to a nonlinear model for the propagation of precipitation fronts, with the front locations being exposed in the sizes of the localized wavelet coefficients. (C) 2011 Elsevier Ltd. All rights reserved.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNP