A multi-level spectral deferred correction method

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.

[1]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[2]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[3]  Edmond Chow,et al.  A Survey of Parallelization Techniques for Multigrid Solvers , 2006, Parallel Processing for Scientific Computing.

[4]  Andrew J. Christlieb,et al.  Integral deferred correction methods constructed with high order Runge-Kutta integrators , 2009, Math. Comput..

[5]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[6]  Rolf Krause,et al.  Inexact spectral deferred corrections , 2016 .

[7]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[8]  Rolf Krause,et al.  A massively space-time parallel N-body solver , 2012, 2012 International Conference for High Performance Computing, Networking, Storage and Analysis.

[9]  Jingfang Huang,et al.  Accelerating the convergence of spectral deferred correction methods , 2006, J. Comput. Phys..

[10]  Yong Jung Kim.  A MATHEMATICAL INTRODUCTION TO FLUID MECHANICS , 2008 .

[11]  Michael L. Minion,et al.  TOWARD AN EFFICIENT PARALLEL IN TIME METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS , 2012 .

[12]  Michael L. Minion,et al.  A HYBRID PARAREAL SPECTRAL DEFERRED CORRECTIONS METHOD , 2010 .

[13]  Anders C. Hansen,et al.  Convergence , 1992, Journal of Political Economy.

[14]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[15]  L. Greengard,et al.  Spectral Deferred Correction Methods for Ordinary Differential Equations , 2000 .

[16]  Larry L. Schumaker,et al.  Interated Deferred Corrections for Initial Value Problems , 1967 .

[17]  Martin Weiser,et al.  Faster SDC convergence on non-equidistant grids by DIRK sweeps , 2015 .

[18]  Michael L. Minion,et al.  Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics , 2004 .

[19]  Rolf Krause,et al.  Explicit Parallel-in-time Integration of a Linear Acoustic-Advection System , 2012, ArXiv.

[20]  Benjamin W. Ong,et al.  Semi-implicit integral deferred correction constructed with additive Runge-Kutta methods , 2011 .

[21]  Benjamin W. Ong,et al.  COMMENTS ON HIGH-ORDER INTEGRATORS EMBEDDED WITHIN INTEGRAL DEFERRED CORRECTION METHODS , 2009 .

[22]  Hans J. Stetter.  Economical Global Error Estimation , 1974 .

[23]  Anita T. Layton,et al.  On the efficiency of spectral deferred correction methods for time-dependent partial differential equations , 2009 .

[24]  V. Pereyra.  On improving an approximate solution of a functional equation by deferred corrections , 1966 .

[25]  Rolf Krause,et al.  Inexact spectral deferred corrections using single-cycle multigrid , 2014 .

[26]  Graham F. Carey,et al.  A high-order compact formulation for the 3D Poisson equation , 1996 .

[27]  Michael L. Minion,et al.  Semi-implicit projection methods for incompressible flow based on spectral deferred corrections , 2004 .

[28]  A. Bourlioux,et al.  High-order multi-implicit spectral deferred correction methods for problems of reactive flow , 2003 .

[29]  Michael L. Minion,et al.  Efficient Implementation of a Multi-Level Parallel in Time Algorithm , 2014 .

[30]  Thomas Hagstrom,et al.  On the spectral deferred correction of splitting methods for initial value problems , 2006 .

[31]  Hans J. Stetter,et al.  The Defect Correction Approach , 1984 .

[32]  Victor Pereyra.  Iterated deferred corrections for nonlinear operator equations , 1967 .

[33]  Michael L. Minion,et al.  Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations , 2005 .

[34]  Achi Brandt,et al.  Application of a multi-level grid method to transonic flow calculations , 1976 .

[35]  Elizabeth L. Bouzarth,et al.  A multirate time integrator for regularized Stokeslets , 2010, J. Comput. Phys..

[36]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[37]  Marco Donatelli,et al.  A V-cycle Multigrid for multilevel matrix algebras: proof of optimality , 2007, Numerische Mathematik.

[38]  Geōrgios Iōannou Kontopoulos,et al.  The theory of orbits in the solar system and in stellar systems, organized in cooperation with Cospar, held in Thessaloniki, Greece, 17-22 August 1964 , 1966 .

[39]  M. Minion.  Semi-implicit spectral deferred correction methods for ordinary differential equations , 2003 .

[40]  Yinhua Xia,et al.  Efficient time discretization for local discontinuous Galerkin methods , 2007 .

[41]  Uri M. Ascher,et al.  Ordinary Differential Equations , 1998 .

[42]  John D. Hunter,et al.  Matplotlib: A 2D Graphics Environment , 2007, Computing in Science & Engineering.

[43]  Terry Haut,et al.  An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs , 2013, SIAM J. Sci. Comput..

[44]  P. E. Zadunaisky.  A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations , 1966 .

[45]  Oleg V. Vasilyev,et al.  Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations , 2006, J. Comput. Phys..

[46]  Y. Maday,et al.  Symmetric parareal algorithms for Hamiltonian systems , 2010, 1011.6222.