A class of hybrid linear multistep methods with A(ɑ)-stability properties for stiff IVPs in ODEs

Abstract In this paper, we consider a family of hybrid linear multistep methods (LMM) with reasonable error order for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A(ɑ)-stable for k = 1, ...,9.

[1]  C. Führer,et al.  A collocation formulation of multistep methods for variable step-size extensions , 2002 .

[2]  Zarina Bibi Ibrahim,et al.  Variable Step Block Backward Differentiation Formula for Solving First Order Stiff ODEs , 2007, World Congress on Engineering.

[3]  M. Ikhile,et al.  A Continuous Formulation of A(α)-Stable Second Derivative Linear Multistep Methods for Stiff IVPs in ODEs , 2012 .

[4]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[5]  Ernst Hairer,et al.  Rosenbrock-Type Methods , 1996 .

[6]  John C. Butcher,et al.  Second derivative methods with RK stability , 2005, Numerical Algorithms.

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

[8]  C. W. Gear,et al.  The automatic integration of stiff ordinary differential equations , 1968, IFIP Congress.

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

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

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

[12]  W. Enright.  Continuous numerical methods for ODEs with defect control , 2000 .

[13]  Tianhai Tian,et al.  Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations , 2001 .

[14]  M. Ikhile,et al.  A ($\alpha $)-Stable Linear Multistep Methods for Stiff IVPs in ODEs , 2011 .

[15]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[16]  John C. Butcher,et al.  Trees and numerical methods for ordinary differential equations , 2010, Numerical Algorithms.

[17]  J. Butcher.  Forty‐five years of A‐stability , 2008 .

[18]  J. C. Butcher.  High order A-stable numerical methods for stiff problems , 2005 .

[19]  David A. Voss,et al.  Fourth-order parallel rosenbrock formulae for stiff systems , 2004, Math. Comput. Model..

[20]  Nicholas J. Higham,et al.  Matlab guide , 2000 .

[21]  Olof B. Widlund,et al.  A note on unconditionally stable linear multistep methods , 1967 .

[22]  W. H. Enright,et al.  Second Derivative Multistep Methods for Stiff Ordinary Differential Equations , 1974 .