Error propagation of general linear methods for ordinary differential equations

We discuss error propagation for general linear methods for ordinary differential equations up to terms of order p+2, where p is the order of the method. These results are then applied to the estimation of local discretization errors for methods of order p and for the adjacent order p+1. The results of numerical experiments confirm the reliability of these estimates. This research has applications in the design of robust stepsize and order changing strategies for algorithms based on general linear methods.

[1]  Minoru Urabe,et al.  On Numerical Integration of Ordinary Differential Equations , 1953 .

[2]  Urs Kirchgraber,et al.  Multi-step methods are essentially one-step methods , 1986 .

[3]  P. Chartier,et al.  The potential of parallel multi-value methods for the simulation of large real-life problems , 1998 .

[4]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[5]  Helmut Podhaisky,et al.  On error estimation in general linear methods for stiff ODEs , 2006 .

[6]  Zdzislaw Jackiewicz,et al.  Unconditionally Stable General Linear Methods for Ordinary Differential Equations , 2004 .

[7]  Z. Jackiewicz.  Construction and implementation of general linear methods for ordinary differential equations: A review , 2005 .

[8]  S. Yakowitz,et al.  The Numerical Solution of Ordinary Differential Equations , 1978 .

[9]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[10]  John C. Butcher,et al.  Diagonally-implicit multi-stage integration methods , 1993 .

[11]  Zdzislaw Jackiewicz,et al.  Nordsieck representation of DIMSIMs , 2004, Numerical Algorithms.

[12]  Zdzislaw Jackiewicz,et al.  Construction of General Linear Methods with Runge–Kutta Stability Properties , 2004, Numerical Algorithms.

[13]  John C. Butcher,et al.  The Construction of Practical General Linear Methods , 2003 .

[14]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[15]  Zdzislaw Jackiewicz,et al.  Implementation of DIMSIMs for stiff differential systems , 2002 .

[16]  L. Shampine,et al.  Numerical Solution of Ordinary Differential Equations. , 1995 .

[17]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[18]  Zdzislaw Jackiewicz,et al.  A new approach to error estimation for general linear methods , 2003, Numerische Mathematik.

[19]  Zdzislaw Jackiewicz,et al.  Experiments with a variable-order type 1 DIMSIM code , 1999, Numerical Algorithms.

[20]  Zdzislaw Jackiewicz,et al.  Construction and Implementation of General Linear Methods for Ordinary Differential Equations: A Review , 2005, J. Sci. Comput..

[21]  Nicola Guglielmi,et al.  On the asymptotic properties of a family of matrices , 2001 .

[22]  Zdzislaw Jackiewicz,et al.  Accurate Implicit–Explicit General Linear Methods with Inherent Runge–Kutta Stability , 2009, Journal of Scientific Computing.

[23]  Daniel Stoffer,et al.  General linear methods: connection to one step methods and invariant curves , 1993 .

[24]  Lawrence F. Shampine,et al.  Solving ODEs with MATLAB , 2002 .

[25]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[26]  Nicola Guglielmi,et al.  On the zero-stability of variable stepsize multistep methods: the spectral radius approach , 2001, Numerische Mathematik.