Parallel exponential Rosenbrock methods

Exponential Rosenbrock integrators were shown to be very efficient in solving large systems of stiff ordinary differential equations. So far, such exponential methods have been derived up to order 5. The aim of this paper is to construct new integrators of orders 4, 5, and 6. In contrast to the existing schemes, the new schemes, which are called parallel exponential Rosenbrock integrators, can be implemented on a multi-processor system or parallel computers. The new schemes of orders 4 and 5 require the same number of stages as the old schemes of the same orders of accuracy. However, while the parallel integrator of order 4 can be implemented with the same cost as a 2-stage method, the ones of orders 5 and 6 can be implemented at the cost of a 3-stage method only. This offers a significant improvement over the old schemes in terms of computational time when implemented in parallel. The numerical experiments show the efficiency of the new integrators as well as the comparative performance with the old ones.

[1]  N. Cong,et al.  Parallel iteration of symmetric Runge-Kutta methods for nonstiff initial-value problems , 1994 .

[2]  Kevin Burrage,et al.  Parallel iterated methods based on multistep Runge-Kutta methods of Radau type , 1997, Adv. Comput. Math..

[3]  Marlis Hochbruck,et al.  Exponential Integrators for Large Systems of Differential Equations , 1998, SIAM J. Sci. Comput..

[4]  Vu Thai Luan,et al.  Explicit exponential Runge-Kutta methods of high order for parabolic problems , 2013, J. Comput. Appl. Math..

[5]  Pierluigi Amodio,et al.  Parallel solution in time of ODEs: some achievements and perspectives , 2009 .

[6]  A. Iserles,et al.  On the Theory of Parallel Runge—Kutta Methods , 1990 .

[7]  C. W. Gear,et al.  Parallel methods for ordinary differential equations , 1988 .

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

[9]  David A. Pope.  An exponential method of numerical integration of ordinary differential equations , 1963, CACM.

[10]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[11]  B. P. Sommeijer.  Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations , 1991 .

[12]  A. Ostermann,et al.  Implementation of exponential Rosenbrock-type integrators , 2009 .

[13]  Mayya Tokman,et al.  Implementation of Parallel Adaptive-Krylov Exponential Solvers for Stiff Problems , 2014, SIAM J. Sci. Comput..

[14]  Ben P. Sommeijer,et al.  Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems , 1995 .

[15]  Helmut Podhaisky,et al.  A Class of Explicit Two-Step Runge-Kutta Methods with Enlarged Stability Regions for Parallel Computers , 1999, ACPC.

[16]  Marco Vianello,et al.  A Parallel Exponential Integrator for Large-Scale Discretizations of Advection-Diffusion Models , 2005, PVM/MPI.

[17]  Marlis Hochbruck,et al.  Exponential Rosenbrock-Type Methods , 2008, SIAM J. Numer. Anal..

[18]  Marlis Hochbruck,et al.  Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems , 2005, SIAM J. Numer. Anal..

[19]  Awad H. Al-Mohy,et al.  Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators , 2011, SIAM J. Sci. Comput..

[20]  Vu Thai Luan,et al.  Exponential Rosenbrock methods of order five - construction, analysis and numerical comparisons , 2014, J. Comput. Appl. Math..

[21]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..

[22]  Vu Thai Luan,et al.  Exponential B-Series: The Stiff Case , 2013, SIAM J. Numer. Anal..

[23]  Kevin Burrage,et al.  Parallel methods for initial value problems , 1993 .

[24]  Jürg Nievergelt,et al.  Parallel methods for integrating ordinary differential equations , 1964, CACM.

[25]  Martin J. Gander,et al.  Analysis of the Parareal Time-Parallel Time-Integration Method , 2007, SIAM J. Sci. Comput..

[26]  Mayya Tokman,et al.  Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods , 2006, J. Comput. Phys..

[27]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .