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Algorithm 965: RIDC methods: a family of parallel time integrators. (English) Zbl 1369.65084

MSC:
65L05 Numerical methods for initial value problems
65Y05 Parallel numerical computation
65Y15 Packaged methods for numerical algorithms
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[1] S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, et al. 2014. PETSc Users Manual Revision 3.7. Retrieved July 17, 2016, from http://www.mcs.anl.gov/petsc/petsc-current/docs/manual.pdf.
[2] Peter N. Brown, George D. Byrne, and Alan C. Hindmarsh. 1989. VODE: A variable-coefficient ODE solver. SIAM Journal on Scientific Computing 10, 5, 1038–1051. DOI:http://dx.doi.org/10.1137/0910062 · Zbl 0677.65075 · doi:10.1137/0910062
[3] Kevin Burrage. 1993. Parallel methods for initial value problems. Applied Numerical Mathematics 11, 1–3, 5–25. DOI:http://dx.doi.org/10.1016/0168-9274(93)90037-R · Zbl 0781.65060 · doi:10.1016/0168-9274(93)90037-R
[4] Andrew Christlieb, Ronald Haynes, and Benjamin Ong. 2012. A parallel space-time algorithm. SIAM Journal on Scientific Computing 34, 5, 233–248. · Zbl 1259.65143 · doi:10.1137/110843484
[5] Andrew Christlieb, Colin MacDonald, Benjamin Ong, and Raymond Spiteri. 2015. Revisionist integral deferred correction with adaptive step-size control. Communications in Applied Mathematics and Computational Science 10, 1, 1–25. DOI:http://dx.doi.org/10.2140/camcos.2015.10.1 · Zbl 1312.65110 · doi:10.2140/camcos.2015.10.1
[6] Andrew Christlieb and Benjamin Ong. 2011. Implicit parallel time integrators. Journal of Scientific Computing 49, 2, 167–179. DOI:http://dx.doi.org/10.1007/s10915-010-9452-4 · Zbl 1243.65076 · doi:10.1007/s10915-010-9452-4
[7] Andrew Christlieb, Benjamin Ong, and Jing-Mei Qiu. 2009. Comments on high-order integrators embedded within integral deferred correction methods. Communications in Applied Mathematics and Computational Science 4, 27–56. DOI:http://dx.doi.org/10.2140/camcos.2009.4.27 · Zbl 1167.65389 · doi:10.2140/camcos.2009.4.27
[8] Andrew Christlieb, Benjamin Ong, and Jing-Mei Qiu. 2010a. Integral deferred correction methods constructed with high order Runge-Kutta integrators. Mathematics of Computation 79, 270, 761–783. DOI:http://dx.doi.org/10.1090/S0025-5718-09-02276-5 · Zbl 1209.65073 · doi:10.1090/S0025-5718-09-02276-5
[9] Andrew J. Christlieb, Colin B. MacDonald, and Benjamin W. Ong. 2010b. Parallel high-order integrators. SIAM Journal on Scientific Computing 32, 2, 818–835. DOI:http://dx.doi.org/10.1137/09075740X · Zbl 1211.65089 · doi:10.1137/09075740X
[10] Alok Dutt, Leslie Greengard, and Vladimir Rokhlin. 2000. Spectral deferred correction methods for ordinary differential equations. BIT Numerical Mathematics 40, 2, 241–266. · Zbl 0959.65084 · doi:10.1023/A:1022338906936
[11] Wael Elwasif, Samantha Foley, David Bernholdt, Lee Berry, Debasmita Samaddar, David Newman, and Raul Sanchez. 2011. A dependency-driven formulation of parareal: Parallel-in-time solution of PDEs as a many-task application. In Proceedings of the 2011 ACM International Workshop on Many Task Computing on Grids and Supercomputers (MTAGS’11). ACM, New York, NY, 15–24. DOI:http://dx.doi.org/10.1145/2132876.2132883 · doi:10.1145/2132876.2132883
[12] Matthew Emmett. 2013. PyPFASST: Parallel Full Approximation Scheme in Space and Time. Retrieved July 17, 2016, from http://pypfasst.readthedocs.org/en/latest.
[13] Matthew Emmett, Torbjorn Klatt, Robert Speck, and Daniel Ruprecht. 2015. Parallel Full Approximation Scheme in Space and Time. Retrieved July 17, 2016, from https://github.com/Parallel-in-Time/PFASST.
[14] Matthew Emmett and Michael L. Minion. 2012. Toward an efficient parallel in time method for partial differential equations. Communications in Applied Mathematics and Computational Science 7, 1, 105–132. · Zbl 1248.65106 · doi:10.2140/camcos.2012.7.105
[15] Matthew Emmett and Michael L. Minion. 2014. Efficient implementation of a multi-level parallel in time algorithm. In Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, Vol. 98. Springer, 359–366. DOI:http://dx.doi.org/10.1007/978-3-319-05789-7_33 · Zbl 1382.65194 · doi:10.1007/978-3-319-05789-7_33
[16] Robert F. Enenkel and Kenneth R. Jackson. 1997. DIMSEMs-diagonally implicit single-eigenvalue methods for the numerical solution of stiff ODEs on parallel computers. Advances in Computational Mathematics 7, 1–2, 97–133. DOI:http://dx.doi.org/10.1023/A:1018986500842 Parallel methods for ODEs. · Zbl 0887.65077 · doi:10.1023/A:1018986500842
[17] R. D. Falgout, S. Friedhoff, T. V. Kolev, S. P. MacLachlan, and J. B. Schroder. 2014. Parallel time integration with multigrid. SIAM Journal on Scientific Computing 36, 6, C635–C661. DOI:http://dx.doi.org/10.1137/130944230 · Zbl 1310.65115 · doi:10.1137/130944230
[18] Martin J. Gander. 2015. 50 years of time parallel time integration. In Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, Vol. 9. Springer, 69–113. DOI:http://dx.doi.org/10.1007/978-3-319-23321-5_3 · Zbl 1337.65127 · doi:10.1007/978-3-319-23321-5_3
[19] Martin J. Gander and Stefan Vandewalle. 2007. On the superlinear and linear convergence of the parareal algorithm. In Domain Decomposition Methods in Science and Engineering XVI. Lecture Notes in Computational Science and Engineering, Vol. 55. Springer, 291–298. DOI:http://dx.doi.org/10.1007/978-3-540-34469-8_34 · doi:10.1007/978-3-540-34469-8_34
[20] Brian Gough. 2009. GNU Scientific Library Reference Manual. Network Theory Ltd.
[21] Ronald D. Haynes and Benjamin W. Ong. 2014. MPI–OpenMP algorithms for the parallel space–time solution of time dependent PDEs. In Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, Vol. 98. Springer, 179–187. DOI:http://dx.doi.org/10.1007/978-3-319-05789-7_14 · Zbl 1382.65243 · doi:10.1007/978-3-319-05789-7_14
[22] Alan C. Hindmarsh. 1983. ODEPACK, a systematized collection of ODE solvers. In Scientific Computing. IMACS, New Brunswick, NJ, 55–64.
[23] Alan C. Hindmarsh, Peter N. Brown, Keith E. Grant, Steven L. Lee, Radu Serban, Dan E. Shumaker, and Carol S. Woodward. 2005. SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. ACM Transactions on Mathematical Software 31, 3, 363–396. DOI:http://dx.doi.org/10.1145/1089014.1089020 · Zbl 1136.65329 · doi:10.1145/1089014.1089020
[24] M. Kappeller, M. Kiehl, M. Perzl, and M. Lenke. 1996. Optimized extrapolation methods for parallel solution of IVPs on different computer architectures. Applied Mathematics and Computation 77, 23, 301–315. DOI:http://dx.doi.org/10.1016/S0096-3003(95)00219-7 · Zbl 0859.65070 · doi:10.1016/S0096-3003(95)00219-7
[25] David I. Ketcheson and Umair Bin Waheed. 2014. A comparison of high-order explicit Runge-Kutta, extrapolation, and deferred correction methods in serial and parallel. Communications in Applied Mathematics and Computational Science 9, 2, 175–200. DOI:http://dx.doi.org/10.2140/camcos.2014.9.175 · Zbl 1314.65102 · doi:10.2140/camcos.2014.9.175
[26] Yvon Maday and Gabriel Turinici. 2002. A parareal in time procedure for the control of partial differential equations. Comptes Rendus Mathematique 335, 4, 387–392. DOI:http://dx.doi.org/ 10.1016/S1631-073X(02)02467-6 · Zbl 1006.65071 · doi:10.1016/S1631-073X(02)02467-6
[27] Willard Miranker and Werner Liniger. 1967. Parallel methods for the numerical integration of ordinary differential equations. Mathematics of Computation 21, 303–320. · Zbl 0155.47204 · doi:10.1090/S0025-5718-1967-0223106-8
[28] Benjamin Ong, Andrew Christlieb, and Andrew Melfi. 2012. Parallel Semi-Implicit Time Integrators. Technical Report. Michigan State University, East Lansing, MI. http://arxiv.org/pdf/1209.4297.pdf. · Zbl 1243.65076
[29] Linda Petzold. 1983. A description of DASSL: A differential/algebraic system solver. In Scientific Computing. IMACS, New Brunswick, NJ, 65–68.
[30] Bernhard Schmitt. 2013. Peer Methods for Ordinary Differential Equations. Retrieved July 17, 2016, from http://www.mathematik.uni-marburg.de/∼schmitt/peer/.
[31] Jacob Schroder, Robert Falgout, Tzanio Kolev, Ulrike Yang, Anders Petersson, Veselin Dobrev, Scott MacLachlan, Stephanie Friedhoff, and Ben O’Neil. 2015. XBraid: Parallel Time Integration with Multigrid. Retrieved July 17, 2016, from http://llnl.gov/casc/xbraid. (2015).
[32] Lawrence Shampine, Mark Reichelt, and Jacek Kierzenka. 1999. Solving index-1 DAEs in MATLAB and Simulink. SIAM Review 41, 3, 538–552. DOI:http://dx.doi.org/10.1137/S003614459933425X · Zbl 0935.65082 · doi:10.1137/S003614459933425X
[33] Walter Vandevender and Karen Haskell. 1982. The SLATEC mathematical subroutine library. ACM SIGNUM Newsletter 17, 3, 16–21. · doi:10.1145/1057594.1057595
[34] Stefan Vandewalle and Dirk Roose. 1989. The parallel waveform relaxation multigrid method. In Proceedings of the 3rd SIAM Conference on Parallel Processing for Scientific Computing. 152–156.
[35] Rüdiger Weiner, Katja Biermann, Bernhard A. Schmitt, and Helmut Podhaisky. 2008. Explicit two-step peer methods. Computers and Mathematics with Applications 55, 4, 609–619. DOI:http://dx.doi.org/ 10.1016/j.camwa.2007.04.026 · Zbl 1142.65060 · doi:10.1016/j.camwa.2007.04.026
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