Relaxed Multirate Infinitesimal Step Methods

This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate version of the Multirate Infinitesimal Step methods, and use recent theoretical developments for Generalized Additive Runge-Kutta methods to propose our higher-order Relaxed Multirate Infinitesimal Step extensions. The resulting framework supports a range of attractive properties for multirate methods, including telescopic extensions, subcycling, embeddings for temporal error estimation, and support for changes to the fast/slow time-scale separation between steps, without requiring any sacrifices in linear stability. In addition to providing rigorous theoretical developments for these new methods, we provide numerical tests demonstrating convergence and efficiency on a suite of multirate test problems.

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

[2]  I. Prigogine,et al.  On symmetry-breaking instabilities in dissipative systems , 1967 .

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

[4]  Oswald Knoth,et al.  Generalized Split-Explicit Runge–Kutta Methods for the Compressible Euler Equations , 2014 .

[5]  E. Hairer,et al.  Dense output for extrapolation methods , 1990 .

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

[7]  Adrian Sandu,et al.  Multirate generalized additive Runge Kutta methods , 2016, Numerische Mathematik.

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

[9]  Jean-François Remacle,et al.  Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows , 2013 .

[10]  Francis X. Giraldo,et al.  A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases , 2008, J. Comput. Phys..

[11]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[12]  P. Rentrop,et al.  Multirate Partitioned Runge-Kutta Methods , 2001 .

[13]  Jorge F. Oliveira,et al.  Radio Frequency Numerical Simulation Techniques Based on Multirate Runge-Kutta Schemes , 2012, J. Appl. Math..

[14]  Ralf Wolke,et al.  Implicit-explicit Runge-Kutta methods for computing atmospheric reactive flows , 1998 .

[15]  Adrian Sandu,et al.  A Generalized-Structure Approach to Additive Runge-Kutta Methods , 2015, SIAM J. Numer. Anal..

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

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

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

[19]  Michael Günther,et al.  Hierarchical Mixed Multirating in Circuit Simulation , 2007 .

[20]  Adrian Sandu.  A Class of Multirate Infinitesimal GARK Methods , 2019, SIAM J. Numer. Anal..

[21]  Adrian Sandu,et al.  A class of generalized additive Runge-Kutta methods , 2013, ArXiv.

[22]  Valeriu Savcenco.  Construction of a multirate RODAS method for stiff ODEs , 2009 .

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

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

[25]  J. Klemp,et al.  The Simulation of Three-Dimensional Convective Storm Dynamics , 1978 .

[26]  M. Schlegel,et al.  Implementation of multirate time integration methods for air pollution modelling , 2012 .

[27]  Ralf Wolke,et al.  Multirate Runge-Kutta schemes for advection equations , 2009 .

[28]  Jens Lang,et al.  Comparison of the asymptotic stability for multirate Rosenbrock methods , 2014, J. Comput. Appl. Math..

[29]  Emil M. Constantinescu,et al.  Multiphysics simulations , 2013, HiPC 2013.

[30]  A Arie Verhoeven,et al.  Redundancy reduction of IC models : by multirate time-integration and model order reduction , 2008 .

[31]  Adrian Sandu,et al.  Design of High-Order Decoupled Multirate GARK Schemes , 2018, SIAM J. Sci. Comput..

[32]  Valeriu Savcenco.  Comparison of the asymptotic stability properties for two multirate strategies , 2007 .

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

[34]  Oswald Knoth,et al.  Multirate infinitesimal step methods for atmospheric flow simulation , 2009 .

[35]  Charles William Gear.  Multirate methods for ordinary differential equations , 1974 .

[36]  Pak-Wing Fok,et al.  A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control , 2016, J. Sci. Comput..

[37]  Sascha Bremicker-Trübelhorn,et al.  On Multirate GARK Schemes with Adaptive Micro Step Sizes for Fluid–Structure Interaction: Order Conditions and Preservation of the Geometric Conservation Law , 2017 .

[38]  W. Skamarock,et al.  The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations , 1992 .

[39]  Willem Hundsdorfer,et al.  A multirate time stepping strategy for stiff ordinary differential equations , 2007 .

[40]  Adrian Sandu,et al.  Extrapolated Multirate Methods for Differential Equations with Multiple Time Scales , 2013, J. Sci. Comput..

[41]  Ralf Wolke,et al.  Numerical solution of multiscale problems in atmospheric modeling , 2012 .

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

[43]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[44]  R.M.M. Mattheij,et al.  Multirate numerical integration for stiff ODEs , 2010 .