On the derivation of explicit two-step peer methods

The so-called two-step peer methods for the numerical solution of Initial Value Problems (IVP) in differential systems were introduced by R. Weiner, B.A. Schmitt and coworkers as a tool to solve different types of IVPs either in sequential or parallel computers. These methods combine the advantages of Runge-Kutta (RK) and multistep methods to obtain high stage order and therefore provide in a natural way a dense output. In particular, several explicit peer methods have been proved to be competitive with standard RK methods in a wide selection of non-stiff test problems. The aim of this paper is to propose an alternative procedure to construct families of explicit two step peer methods in which the available parameters appear in a transparent way. This allows us to obtain families of fixed stepsize s stage methods with stage order 2s-1, which provide dense output without extra cost, depending on some free parameters that can be selected taking into account the stability properties and leading error terms. A study of the extension of these methods to variable stepsize is also carried out. Optimal s stage methods with s=2,3 are derived.

[1]  Rüdiger Weiner,et al.  Parameter optimization for explicit parallel peer two-step methods , 2009 .

[2]  Rüdiger Weiner,et al.  Explicit Two-Step Peer Methods for the Compressible Euler Equations , 2009 .

[3]  L. Shampine,et al.  A 3(2) pair of Runge - Kutta formulas , 1989 .

[4]  E. Hairer,et al.  Solving Ordinary ,Differential Equations I, Nonstiff problems/E. Hairer, S. P. Norsett, G. Wanner, Second Revised Edition with 135 Figures, Vol.: 1 , 2000 .

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

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

[7]  Helmut Podhaisky,et al.  Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer , 1998 .

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

[9]  Rüdiger Weiner,et al.  Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation , 2010, J. Comput. Appl. Math..

[10]  Rüdiger Weiner,et al.  Implicit parallel peer methods for stiff initial value problems , 2005 .

[11]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[12]  Rüdiger Weiner,et al.  Parallel Two-Step W-Methods with Peer Variables , 2004, SIAM J. Numer. Anal..

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

[14]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[15]  Zdzislaw Jackiewicz,et al.  Variable-stepsize explicit two-step Runge-Kutta methods , 1992 .

[16]  Helmut Podhaisky,et al.  Superconvergent explicit two-step peer methods , 2009 .

[17]  J. Butcher.  The Numerical Analysis of Ordinary Di erential Equa-tions , 1986 .

[18]  Zdzislaw Jackiewicz,et al.  A general class of two-step Runge-Kutta methods for ordinary differential equations , 1995 .

[19]  Bruno Welfert,et al.  Two-Step Runge-Kutta: Theory and Practice , 2000 .

[20]  Helmut Podhaisky,et al.  Explicit two-step peer methods , 2008, Comput. Math. Appl..

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

[22]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[23]  Helmut Podhaisky,et al.  Linearly-implicit two-step methods and their implementation in Nordsieck form , 2006 .

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