A Hybrid Implicit-Explicit Adaptive Multirate Numerical Scheme for Time-Dependent Equations

We develop a hybrid implicit and explicit adaptive multirate time integration method to solve systems of time-dependent equations that present two significantly different scales. We adopt an iteration scheme to decouple the equations with different time scales. At each iteration, we use an implicit Galerkin method with a fast time-step to solve for the fast scale variables and an explicit method with a slow time-step to solve for the slow variables. We derive an error estimator using a posteriori analysis which controls both the iteration number and the adaptive time-step selection. We present several numerical examples demonstrating the efficiency of our scheme and conclude with a stability analysis for a model problem.

[1]  Robert L. Higdon,et al.  Barotropic-Baroclinic Time Splitting for Ocean Circulation Modeling , 1997 .

[2]  Anders Logg,et al.  Multi-Adaptive Galerkin Methods for ODEs I , 2002, SIAM J. Sci. Comput..

[3]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[4]  Emil M. Constantinescu,et al.  Multirate Explicit Adams Methods for Time Integration of Conservation Laws , 2009, J. Sci. Comput..

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

[6]  Andreas Bartel,et al.  A multirate W-method for electrical networks in state-space formulation , 2002 .

[7]  Stig Skelboe,et al.  Stability of backward Euler multirate methods and convergence of waveform relaxation , 1992 .

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

[9]  Willem Hundsdorfer,et al.  Analysis of a multirate theta-method for stiff ODEs , 2009 .

[10]  T. Dupont,et al.  A Finite Difference Domain Decomposition Algorithm for Numerical Solution of the Heat Equation , 1989 .

[11]  Christian Lubich,et al.  MUR8: a multirate extension of the eighth-order Dormand-Prince method , 1997 .

[12]  J. Dukowicz,et al.  Implicit free‐surface method for the Bryan‐Cox‐Semtner ocean model , 1994 .

[13]  Amir Averbuch,et al.  Implicit-Explicit Parallel Asynchronous Solver of Parabolic PDEs , 1998, SIAM J. Sci. Comput..

[14]  J. F. Andrus,et al.  Numerical Solution of Systems of Ordinary Differential Equations Separated into Subsystems , 1979 .

[15]  Robert D. Skeel,et al.  Dangers of multiple time step methods , 1993 .

[16]  C. W. Gear,et al.  Multirate linear multistep methods , 1984 .

[17]  John N. Shadid,et al.  An A Posteriori-A Priori Analysis of Multiscale Operator Splitting , 2008, SIAM J. Numer. Anal..

[18]  Josef M. Oberhuber,et al.  Simulation of the atlantic circulation with a coupled sea ice-mixed layer-isopycnal general circulat , 1993 .

[19]  D. Estep,et al.  A Posteriori analysis of a multirate numerical method for ordinary differential equations , 2012 .

[20]  J. F. Andrus.  Stability of a multi-rate method for numerical integration of ODE's , 1993 .

[21]  E. Hofer,et al.  A Partially Implicit Method for Large Stiff Systems of ODEs with Only Few Equations Introducing Small Time-Constants , 1976 .

[22]  Emil M. Constantinescu,et al.  Multirate Timestepping Methods for Hyperbolic Conservation Laws , 2007, J. Sci. Comput..