Local Linearization - Runge-Kutta methods: A class of A-stable explicit integrators for dynamical systems

Abstract A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge–Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.

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

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

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

[4]  A. Iserles,et al.  On the Implementation of the Method of Magnus Series for Linear Differential Equations , 1999 .

[5]  Tien D. Bui.  Some A-Stable and L-Stable Methods for the Numerical Integration of Stiff Ordinary Differential Equations , 1979, JACM.

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

[7]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

[8]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[9]  Arieh Iserles,et al.  Solving linear ordinary differential equations by exponentials of iterated commutators , 1984 .

[10]  Julyan H. E. Cartwright,et al.  THE DYNAMICS OF RUNGE–KUTTA METHODS , 1992 .

[11]  W. Beyn.  On the Numerical Approximation of Phase Portraits Near Stationary Points , 1987 .

[12]  Juan C. Jiménez,et al.  A higher order local linearization method for solving ordinary differential equations , 2007, Appl. Math. Comput..

[13]  G. R. W. Quispel,et al.  Linearization-preserving self-adjoint and symplectic integrators , 2009 .

[14]  Arieh Iserles.  Quadrature methods for stiff ordinary differential systems , 1981 .

[15]  B. V. Pavlov,et al.  The method of local linearization in the numerical solution of stiff systems of ordinary differential equations , 1988 .

[16]  R. K. Jain.  Some A -Stable Methods for Stiff Ordinary Differential Equations , 1972 .

[17]  C. Loan.  Computing integrals involving the matrix exponential , 1978 .

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

[19]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[20]  J. D. Lawson.  Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants , 1967 .

[21]  Antonella Zanna,et al.  Collocation and Relaxed Collocation for the Fer and the Magnus Expansions , 1999 .

[22]  Arieh Iserles,et al.  $A$-stability and dominating pairs , 1978 .

[23]  Karline Soetaert,et al.  Solving Ordinary Differential Equations in R , 2012 .

[24]  L. Tuckerman,et al.  A method for exponential propagation of large systems of stiff nonlinear differential equations , 1989 .

[25]  Juan I. Ramos,et al.  Piecewise-linearized methods for initial-value problems , 1997 .

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

[27]  Jeff Cash,et al.  On the Exponential Fitting of Composite, Multiderivative Linear Multistep Methods , 1981 .

[28]  A. Iserles,et al.  Methods for the approximation of the matrix exponential in a Lie‐algebraic setting , 1999, math/9904122.

[29]  Ian W. Turner,et al.  Efficient simulation of unsaturated flow using exponential time integration , 2011, Appl. Math. Comput..

[30]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

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

[32]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[33]  Matematik,et al.  Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods , 2005 .

[34]  L. M. Pedroso,et al.  Computing multiple integrals involving matrix exponentials , 2007 .

[35]  Wolf-Jürgen Beyn,et al.  On invariant closed curves for one-step methods , 1987 .

[36]  S. P. Nørsett.  An A-stable modification of the Adams-Bashforth methods , 1969 .

[37]  Nicholas J. Higham,et al.  The Scaling and Squaring Method for the Matrix Exponential Revisited , 2005, SIAM J. Matrix Anal. Appl..

[38]  D. A. Voss.  A fifth-order exponentially fitted formula , 1988 .

[39]  Juan C. Jiménez,et al.  Locally Linearized Runge Kutta method of Dormand and Prince , 2012, Appl. Math. Comput..

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

[41]  Juan Carlos Jimenez,et al.  A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations , 2002, Appl. Math. Lett..

[42]  J. C. Jimenez,et al.  Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise , 2012 .

[43]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[44]  Fernando Casas,et al.  Improved High Order Integrators Based on the Magnus Expansion , 2000 .

[45]  Ian Stewart,et al.  Warning — handle with care! , 1992, Nature.

[46]  R. J. Biscay,et al.  Approximation of continuous time stochastic processes by the local linearization method revisited , 2002 .

[47]  Juan C. Jiménez,et al.  Rate of convergence of local linearization schemes for initial-value problems , 2005, Appl. Math. Comput..

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

[49]  Juan C. Jiménez,et al.  Dynamic properties of the local linearization method for initial-value problems , 2002, Appl. Math. Comput..

[50]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[51]  Alexander Ostermann,et al.  Exponential multistep methods of Adams-type , 2011 .

[52]  Abdus Salam,et al.  LOCAL LINEARIZATION METHODS FOR THE NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS: AN OVERVIEW , 2009 .

[53]  M. Hochbruck,et al.  Exponential Runge--Kutta methods for parabolic problems , 2005 .

[54]  John Carroll.  A Matricial Exponentially Fitted Scheme for the Numerical Solution of Stiff Initial-Value Problems , 1993 .

[55]  Juan C. Jiménez,et al.  Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations , 2006, International Conference on Computational Science.

[56]  Patrick W. Gaffney,et al.  A Performance Evaluation of Some FORTRAN Subroutines for the Solution of Stiff Oscillatory Ordinary Differential Equations , 1984, TOMS.

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

[58]  G. Quispel,et al.  Splitting methods , 2002, Acta Numerica.

[59]  L. Dieci,et al.  Padé approximation for the exponential of a block triangular matrix , 2000 .

[60]  Ralph A. Willoughby,et al.  EFFICIENT INTEGRATION METHODS FOR STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS , 1970 .

[61]  L. M. Pedroso,et al.  Letter to the Editor: Computing multiple integrals involving matrix exponentials , 2008 .