Preconditioned implicit-exponential integrators (IMEXP) for stiff PDEs

Abstract We propose two new classes of time integrators for stiff DEs: the implicit exponential (IMEXP) and the hybrid exponential methods. In contrast to the existing exponential schemes, the new methods offer significant computational advantages when used with preconditioners. Any preconditioner can be used with any of these new schemes. This leads to a broader applicability of exponential methods. The proof of convergence of these integrators and numerical demonstration of their efficiency are presented.

[1]  G. Rainwater,et al.  A new approach to constructing efficient stiffly accurate EPIRK methods , 2016, J. Comput. Phys..

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

[3]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[4]  Fabrizio Bisetti,et al.  Integration of large chemical kinetic mechanisms via exponential methods with Krylov approximations to Jacobian matrix functions , 2012 .

[5]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[6]  Steven J. Ruuth,et al.  Implicit-Explicit Methods for Time-Dependent PDE''s , 1993 .

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

[8]  Stefaan Poedts,et al.  Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas , 2004 .

[9]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

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

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

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

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

[14]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[15]  Vu Thai Luan,et al.  Parallel exponential Rosenbrock methods , 2016, Comput. Math. Appl..

[16]  Vu Thai Luan,et al.  Stiff Order Conditions for Exponential Runge-Kutta Methods of Order Five , 2012, HPSC.

[17]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[18]  Mayya Tokman,et al.  New Adaptive Exponential Propagation Iterative Methods of Runge-Kutta Type , 2012, SIAM J. Sci. Comput..

[19]  Steven J. Ruuth.  Implicit-explicit methods for reaction-diffusion problems in pattern formation , 1995 .

[20]  Willem Hundsdorfer,et al.  Stability of implicit-explicit linear multistep methods , 1997 .

[21]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[22]  Emil M. Constantinescu,et al.  Extrapolated Implicit-Explicit Time Stepping , 2009, SIAM J. Sci. Comput..

[23]  Mayya Tokman,et al.  Magnetohydrodynamic modeling of solar magnetic arcades using exponential propagation methods , 2001 .

[24]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[25]  Mayya Tokman,et al.  Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs , 2013, J. Comput. Appl. Math..

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

[27]  Weidong Zhao,et al.  Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations , 2016, J. Sci. Comput..

[28]  Mayya Tokman,et al.  A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK) , 2011, J. Comput. Phys..

[29]  Mayya Tokman,et al.  A new class of split exponential propagation iterative methods of Runge-Kutta type (sEPIRK) for semilinear systems of ODEs , 2014, J. Comput. Phys..

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

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

[32]  Philip K. Maini,et al.  Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains , 2007, J. Comput. Phys..

[33]  J. Schnakenberg,et al.  Simple chemical reaction systems with limit cycle behaviour. , 1979, Journal of theoretical biology.

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

[35]  Deborah Lacitignola,et al.  Numerical approximation of Turing patterns in electrodeposition by ADI methods , 2012, J. Comput. Appl. Math..