A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel

Citation A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and We compare the three main types of high-order one-step initial value solvers: extrapolation, spectral deferred correction, and embedded Runge–Kutta pairs. We consider orders four through twelve, including both serial and parallel implementations. We cast extrapolation and deferred correction methods as fixed-order Runge–Kutta methods, providing a natural framework for the comparison. The stability and accuracy properties of the methods are analyzed by theoretical measures, and these are compared with the results of numerical tests. In serial, the eighth-order pair of Prince and Dormand (DOP8) is most efficient. But other high-order methods can be more efficient than DOP8 when implemented in parallel. This is demonstrated by comparing a parallelized version of the well-known ODEX code with the (serial) DOP853 code. For an N-body problem with N = 400, the experimental extrapolation code is as fast as the tuned Runge–Kutta pair at loose tolerances, and is up to two times as fast at tight tolerances.

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

[2]  Thomas Rauber,et al.  Load balancing schemes for extrapolation methods , 1997 .

[3]  Colin B. Macdonald,et al.  Parallel High-Order Integrators , 2010, SIAM J. Sci. Comput..

[4]  M. Kiehl,et al.  Optimized extrapolation methods for parallel solution of IVPs on different computer architectures , 1996 .

[5]  Ben P. Sommeijer,et al.  Parallel ODE solvers , 1990, ICS '90.

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

[7]  D. Tromeur-Dervout,et al.  Cyclic Distribution of Pipelined Parallel Deferred Correction method for ODE/DAE , 2009 .

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

[9]  W..  Solution of Ordinary Differential Initial Value Problems on an Intel Hypercube , 2002 .

[10]  Lawrence F. Shampine,et al.  An efficient Runge-Kutta (4,5) pair , 1996 .

[11]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[12]  R. LeVeque,et al.  Shock dynamics in layered periodic media , 2011, 1105.2892.

[13]  Bertil Gustafsson,et al.  Deferred Correction Methods for Initial Value Problems , 2001 .

[14]  R. Lewis,et al.  Low-storage, Explicit Runge-Kutta Schemes for the Compressible Navier-Stokes Equations , 2000 .

[15]  P. Pair,et al.  An Efficient Runge-Kutta ( 4 , 5 ) , 2003 .

[16]  Stefan Gottschalk.  Parallel And Sequential Methods For Ordinary Differential Equations , 2016 .

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

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

[19]  P. Deuflhard.  Order and stepsize control in extrapolation methods , 1983 .

[20]  T. Fukushima,et al.  Parallelized extrapolation method and its application to the orbital dynamics. , 1997 .

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

[22]  Chi-Wang Shu,et al.  High Order Strong Stability Preserving Time Discretizations , 2009, J. Sci. Comput..

[23]  Manuel Calvo,et al.  A new embedded pair of Runge-Kutta formulas of orders 5 and 6 , 1990 .

[24]  Mengping Zhang,et al.  STRONG STABILITY PRESERVING PROPERTY OF THE DEFERRED CORRECTION TIME DISCRETIZATION , 2008 .

[25]  P. Houwen,et al.  Parallel iteration of high-order Runge-Kutta methods with stepsize control , 1990 .

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

[27]  L. Shampine,et al.  Efficiency comparisons of methods for integrating ODEs , 1994 .

[28]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

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

[30]  J. Dormand,et al.  High order embedded Runge-Kutta formulae , 1981 .

[31]  Peter Deuflhard,et al.  Recent progress in extrapolation methods for ordinary differential equations , 1985 .

[32]  Lawrence F. Shampine,et al.  Fixed versus variable order Runge-Kutta , 1986, TOMS.

[33]  Randall J. LeVeque,et al.  High-Order Wave Propagation Algorithms for Hyperbolic Systems , 2011, SIAM J. Sci. Comput..

[34]  Kenneth R. Jackson,et al.  The potential for parallelism in Runge-Kutta methods. Part 1: RK formulas in standard form , 1995 .

[35]  A. R. Curtis.  High-order Explicit Runge-Kutta Formulae, Their Uses, and Limitations , 1975 .