Spectral Deferred Correction Methods
Spectral Deferred Corrections is an iterative strategy for constructing higher-order temporal evolution schemes for ODEs and PDEs. The iterative nature of SDC methods can be exploited to create higher-order methods for stiff systems in which operator splitting is employed to avoid the computational cost of coupled implicit processes. This multi-implicit (MISDC) strategy is being applied to reacting gas dynamics simulations in which both the reaction and diffusive terms are treated implicitly but independently. Recent work has also considered ways in which these decoupled implicit solves can be done concurrently.
SDC methods are also the key component of the PFASST parallel in time method PFASST. In the PFASST algorithm, multi-level SDC iterations are performed on multiple time slices and in parallel with corrections to the initial solution on each time slice and MLSDC level communicated after each deferred correction sweep.
Other projects underway include evaluating the efficiency of SDC methods for geophysical flows, constructing higher-order projection methods for the incompressible Navier-Stokes equations, and building SDC based Magnus integrators for isospectral flow problems. See below for more project details
☞ For more information, contact Michael Minion firstname.lastname@example.org.
F.P. Hamon, M.S. Day, M.L. Minion, "Concurrent Implicit Spectral Deferred Correction Scheme for Low-Mach Number Combustion with Detailed Chemistry", submitted for publication. [arxiv]
R.W. Grout, H. Kolla, M.L. Minion and J.B. Bell, "Achieving algorithmic resilience for temporal integration through spectral deferred corrections," to appear in Commun. in Appl. Math. and Comput. Sci. [arxiv].
W. Pazner, A. Nonaka, J. B. Bell, M. S. Day, and M. L. Minion, "A High-Order Spectral Deferred Correction Strategy for Low Mach Number Flow With Complex Chemistry," Combust. Theory Modelling, [DOI] , 2016. [pdf]
M. Minion, R. Speck, M. Bolten, M. Emmett, and D. Ruprecht "Interweaving PFASST and parallel multigrid", SIAM J. Scientific Computing, 37(5), pp. S244-S263, 2015.
R. Speck, D. Ruprecht, M. Minion, and R. Krause, "Inexact Spectral Deferred Corrections", Domain Decomposition Methods in Science and Engineering XXII, 104, pp. 127--133, 2015.
R. Speck, D. Ruprecht, M. Emmett, M. Minion, M. Bolten, and R. Krause "A multi-level spectral deferred correction method", BIT Numerical Mathematics, 2014 [arxiv].
M. Emmett and M. Minion "Efficient implementation of a multi-level parallel in time algorithm", Proceedings of the 21st International Conference on Domain Decomposition Methods, DD21, , 98, 359-366, 2014.
A.S. Almgren, A.J. Aspden, J. B. Bell, and M. L. Minion, "On the Use of Higher-Order Projection Methods for Incompressible Turbulent Flow", SIAM J. Sci. Comput., 35, 1, B25-B42, 2013. [pdf]
R. Speck, D. Ruprecht, R. Krause, M. Emmett, M. Minion, M. Winkel, and P. Gibbon, "A massively space-time parallel N-body solver", Proceedings of the International Conference on High Performance Computing, SC'12, 92:1-11, 2012.
R. Speck, D. Ruprecht, R. Krause, M. Emmett, M. Minion, M. Winkel, and P. Gibbon, "Integrating an N-body problem with SDC and PFASST", Proceedings of the 21st International Conference on Domain Decomposition Methods, DD21, 2012.
A. Nonaka, J. B. Bell, M. S. Day, C. Gilet, A. S. Almgren, and M. L. Minion, "A Deferred Correction Coupling Strategy for Low Mach Number Flow with Complex Chemistry", Combustion Theory and Modelling, vol. 16, no. 6, pp. 1053-1088, 2012. [pdf]
M. Emmett and M. Minion "Toward an efficient parallel in time method for partial differential equations", Commun. in Appl. Math. and Comput. Sci, 7(1), 105-132, 2012.