Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.

[1]  H. J. Stetter,et al.  Global error estimation in ODE-solvers , 1978 .

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

[3]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

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

[5]  Helmut Podhaisky,et al.  Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration , 2005 .

[6]  G. Yu. Kulikov,et al.  Global Error Estimation and Extrapolated Multistep Methods For Index 1 Differential-Algebraic Systems , 2005 .

[7]  John C. Butcher,et al.  General linear methods for ordinary differential equations , 2009, Math. Comput. Simul..

[8]  Marinus van Veldhuizen,et al.  D-stability and Kaps-Rentrop-methods , 1984, Computing.

[9]  Rüdiger Weiner,et al.  Global error control in implicit parallel peer methods , 2010 .

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

[11]  Gennady Y. Kulikov,et al.  One-leg variable-coefficient formulas for ordinary differential equations and local–global step size control , 2006, Numerical Algorithms.

[12]  Helmut Podhaisky,et al.  Two-step W-methods and their application to MOL-systems , 2003 .

[13]  G. Yu. Kulikov,et al.  A technique for controlling the global error in multistep methods , 2000 .

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

[15]  R. Skeel.  Thirteen ways to estimate global error , 1986 .

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

[17]  G. Yu. Kulikov,et al.  Local and global error estimation in Nordsieck methods , 2008 .

[18]  Jens Lang,et al.  On Global Error Estimation and Control for Initial Value Problems , 2007, SIAM J. Sci. Comput..

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

[20]  René Aïd,et al.  Numerical investigations on global error estimation for ordinary differential equations , 1997 .

[21]  R. D. Skeel.  Global error estimation and the backward differentiation formulas , 1989 .