Fully Implicit Time Stepping Can Be Efficient on Parallel Computers

Brandon Cloutier, Benson K. Muite, Matteo Parsani

Abstract


Benchmarks in high performance computing often involve a single component used in the full solution of a computational problem, such as the solution of a linear system of equations. In many cases, the choice of algorithm, which can determine the components used, is also important when solving a full problem. Numerical evidence suggests that for the Taylor-Green vortex problem at a Reynolds number of 1600, a second order implicit midpoint rule method can require less computational time than the often used linearly implicit Carpenter-Kennedy method for solving the equations of incompressible fluid dynamics for moderate levels of accuracy at the beginning of the flow evolution. The primary reason is that even though the implicit midpoint rule is fully implicit, it can use a small number of iterations per time step, and thus require less computational work per time step than the Carpenter-Kennedy method. For the same number of timesteps, the Carpenter-Kennedy method is more accurate since it uses a higher order timestepping method.

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References


Adams, M.F., Brown, J., Shalf, J., Straalen, B.V., Strohmaier, E., Williams, S.: HPGMG 1.0: A benchmark for ranking high performance computing systems. Tech. Rep. LBNL 6630E, Lawrence Berkeley National Laboratory, Berkeley, California, USA (2014). https://escholarship.org/uc/item/00r9w79m, accessed: 2019-07-17

Ahrens, J., Geveci B., Law C.: ParaView: An End-User Tool for Large Data Visualization, In the Visualization Handbook. Edited by Hansen C.D. and Johnson C.R., Elsevier (2005). http://www.paraview.org/, accessed: 2019-06-24

Aseeri, S., Muite, B.K., Takahashi, D.: Reproducibility in Benchmarking Parallel Fast Fourier Transform based Applications. In: Companion of the 2019 ACM/SPEC International Conference on Performance Engineering, ICPE '19, 7-11 April 2019, Mumbai, India. pp. 5–8. ACM (2019), DOI: 10.1145/3302541.3313105

Balakrishnan, S., Bargash, A.H., Chen, G., Cloutier, B., Li, N., Malicke, D., Muite, B.K., Quell, M., Rigge, P., San Roman Alerigi, D., Solimani, M., Souza, A., Thiban, A.S., West, J., van Moer, M.: Parallel Spectral Numerical Methods. http://en.wikibooks.org/wiki/Parallel_Spectral_Numerical_Methods, accessed: 2019-06-24

Carpenter, M.H., Kennedy, C.A.: Fourth-Order 2N-Storage Runge-Kutta schemes, NASA Langley Research Center Technical Memorandum 109112. https://ntrs.nasa.gov/search.jsp?R=19940028444 (1994), accessed: 2019-07-17

Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods fundamentals in single domains, Springer (2010), DOI: 10.1007/978-3-540-30726-6

Cenaero: Fifth International Workshop on High-Order CFD Methods. https://how5.cenaero.be/, accessed: 2019-06-24

Childs, H., Brugger, E., Bonnell, K., Meredith, J., Miller, M., Whitlock, B., Max, N.: A contract-based system for large data visualization, IEEE Visualization 2005, 190–198, (2005). https://wci.llnl.gov/codes/visit/, accessed: 2019-06-24

Cloutier, B.: MPI Fortran Implicit Midpoint Rule Incompressible Navier-Stokes Solver. https://github.com/bcloutier/PSNM/blob/master/NavierStokes/Programs/NavierStokes3dFortranMPI/navierstokes_IMR.f90, accessed: 2019-06-24

Cloutier, B.: MPI Fortran Carpenter-Kennedy Incompressible Navier-Stokes Solver. https://github.com/bcloutier/PSNM/blob/master/NavierStokes/Programs/NavierStokes3dFortranMPI/navierstokes.f90, accessed: 2019-06-24

Cloutier, B., Muite, B.K., Parsani, M.: Fully Implicit Time Stepping can be Efficient on Parallel Computers [Data set], Zenodo, DOI: 10.5281/zenodo.2667709

Deutsches Zentrum fur Luft- und Raumfahrt (DLR): Second International Workshop on High-Order CFD Methods. https://www.dlr.de/as/desktopdefault.aspx/tabid-8170/13999_read-35550/, accessed: 2019-06-24

Dongarra, J., Heroux, M., Luszczek, P.: HPCG benchmark: a new metric for ranking high performance computing systems. The International Journal of High Performance Computing Applications 30(1), 3–10 (2015), DOI: 10.1177/1094342015593158

Dongarra, J., Luszczek, P., Petitet, A.: The LINPACK benchmark: Past, present and future. Concurrency and Computation: Practice and Experience 38(9), 803–820 (2003), DOI: 10.1002/cpe.728

Eaton, J. W., Bateman, D., Hauberg, S., Wehbring, R.: GNU Octave version 4.2.1 manual: a high-level interactive language for numerical computations. https://www.gnu.org/software/octave/doc/v4.2.1/ (2017), accessed: 2019-06-24

Elcott, S., Tong, Y., Kanso, E., Schroder, P., Desbrun, M.: Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics 26(1), 4 (2007), DOI: 10.1145/1189762.1189766

Faanes, G., Bataineh, A., Roweth, D., Court, T., Froese, E., Alverson, B., Johnson, T., Kopnick, J., Higgins, M., Reinhard, J.: Cray Cascade: A scalable HPC system based on a Dragonfly network. In: SC'12: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, 1–9 (2012), DOI: 10.1109/SC.2012.39

Hochstleistungsrechenzentrum Stuttgart (HLRS): Hazelhen. https://www.hlrs.de/systems/cray-xc40-hazel-hen/, accessed: 2019-06-24

Hunter, J.D.: Matplotlib: A 2D graphics environment. Computing in Science and Engineering 9(3), 90–95 (2007), DOI: 10.1109/MCSE.2007.55

Ketcheson, D.I., Mortensen, M., Parsani, M., Schilling N.: More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence, arXiv: 1810.10197v1, accessed: 2019-07-17

Li, N., Laizet, S.: 2DECOMP&FFT – A highly scalable 2D decomposition library and FFT interface, Cray User Group 2010, Edinburgh, UK. http://www.2decomp.org/pdf/17B-CUG2010-paper-Ning_Li.pdf, accessed: 2019-07-17

Muller, E.H., Scheichl, R., Vainikko, E.: Petascale solvers for anisotropic PDEs in atmospheric modelling on GPU clusters. Parallel Computing 50, 53–69 (2015), DOI: 10.1016/j.parco.2015.10.007

Parsani, M., Van den Abeele, K., Lacor, C., Turkel, E.: Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Naiver-Stokes equations. Journal of Computational Physics 229(3), 828–850 (2009), DOI: 10.1016/j.jcp.2009.10.014

van Rees, W.M., Leonard, A., Pullin, D.I., Koumoutsakos, P.: A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers. J. of Computational Physics 230, 2794–2805 (2011), DOI: 10.1016/j.jcp.2010.11.031

Yang, C., Xue, W., Fu, H., You, H., Wang, X., Ao, Y., Liu, F., Gan, L., Xu, P., Wang, L., Yang, G., Zheng, W.: 10M-core scalable fully-implicit solver for nonhydrostatic atmospheric dynamics. In: SC'16: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, 57–68 (2016), DOI: 10.1109/SC.2016.5




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