Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations

A new class of stable methods for solving ordinary differential equations (ODEs) is introduced. This is based on combining the Local Linearization (LL) integrator with other extant discretization methods. For this, an auxiliary ODE is solved to determine a correction term that is added to the LL approximation. In particular, combining the LL method with (explicit) Runge Kutta integrators yields what we call LLRK methods. This permits to improve the order of convergence of the LL method without loss of its stability properties. The performance of the proposed integrators is illustrated through computer simulations.

[1]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

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

[3]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

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

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

[6]  J. C. Jimenez,et al.  Local Linearization method for the numerical solution of stochastic differential equations , 1996 .

[7]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[8]  James M. Bower,et al.  The Book of GENESIS , 1994, Springer New York.

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

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

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

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

[13]  K. Burrage,et al.  Numerical methods for strong solutions of stochastic differential equations: an overview , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

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

[16]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

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

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

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

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