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Additional Degrees of Parallelism Within the Adomian Decomposition Method

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Recent Advances in Computational Engineering (ICCE 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 124))

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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 respect 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 than it is with RK.

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Acknowledgements

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 Universität Darmstadt.

This work was partially accomplished during a research stay at NCAR which provided the facilities.

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Authors and Affiliations

  1. TU Darmstadt, Graduate School of Computational Engineering, Darmstadt, Germany

    Andreas Schmitt

  2. TU Darmstadt, Institute of Numerical Methods in Mechanical Engineering, Darmstadt, Germany

    Andreas Schmitt

  3. University of Exeter, Mathematics/Computer Science, Exeter, UK

    Martin Schreiber

  4. Graduate School of Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany

    Michael Schäfer

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  1. Andreas Schmitt
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  2. Martin Schreiber
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  3. Michael Schäfer
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Correspondence to Andreas Schmitt .

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Editors and Affiliations

  1. Graduate School of Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany

    Michael Schäfer

  2. Lehrstuhl für CATS, RWTH Aachen University, Aachen, Germany

    Marek Behr

  3. Institut für Parallele und Verteilte Systeme (IPVS), Universität Stuttgart, Stuttgart, Germany

    Miriam Mehl

  4. Lehrstuhl für Numerische Mathematik, Technische Universität München, Garching bei München, Germany

    Barbara Wohlmuth

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Schmitt, A., Schreiber, M., Schäfer, M. (2018). Additional Degrees of Parallelism Within the Adomian Decomposition Method. In: Schäfer, M., Behr, M., Mehl, M., Wohlmuth, B. (eds) Recent Advances in Computational Engineering. ICCE 2017. Lecture Notes in Computational Science and Engineering, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-93891-2_7

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