Hybrid BDF methods for the numerical solutions of ordinary differential equations

In this article, we have presented the details of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solutions of ordinary differential equations (ODEs). In these hybrid BDF, one additional stage point (or off-step point) has been used in the first derivative of the solution to improve the absolute stability regions. Stability domains of our presented methods have been obtained showing that all these new methods, we say HBDF, of order p, p = 2,4,..., 12, are A(α)-stable whereas they have wide stability regions comparing with those of some known methods like BDF, extended BDF (EBDF), modified EBDF (MEBDF), adaptive EBDF (A-EBDF), and second derivtive Enright methods. Numerical results are also given for five test problems.

[1]  Kevin Burrage,et al.  An implementation of singly-implicit Runge-Kutta methods , 1980 .

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

[3]  Gholamreza Hojjati,et al.  A-EBDF: an adaptive method for numerical solution of stiff systems of ODEs , 2004, Math. Comput. Simul..

[4]  Shoufu Li,et al.  Stability and B-convergence properties of multistep Runge-Kutta methods , 2000, Math. Comput..

[5]  John C. Butcher,et al.  Applications of doubly companion matrices , 2006 .

[6]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[7]  Kevin Burrage,et al.  A special family of Runge-Kutta methods for solving stiff differential equations , 1978 .

[8]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

[9]  C. T. Kelley,et al.  Solving nonlinear equations with Newton's method - fundamentals of algorithms , 2003 .

[10]  John C. Butcher,et al.  DESIRE: diagonally extended singly implicit Runge–Kutta effective order methods , 1998, Numerical Algorithms.

[11]  Jeff Cash,et al.  The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae , 1983 .

[12]  Jeff Cash,et al.  On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae , 1980 .

[13]  G. Psihoyios,et al.  Solving time dependent PDEs via an improved modified extended BDF scheme , 2007, Appl. Math. Comput..

[14]  Roland England.  Some hybrid implicit stiffly stable methods for ordinary differential equations , 1982 .

[15]  Jeff R. Cash,et al.  An MEBDF code for stiff initial value problems , 1992, TOMS.

[16]  John C. Butcher,et al.  DESI methods for stiff initial-value problems , 1996, TOMS.

[17]  J. Cash.  Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs , 2000 .

[18]  J. Cash,et al.  The MOL solution of time dependent partial differential equations , 1996 .

[19]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

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

[21]  Xinyuan Wu,et al.  A sixth-order A-stable explicit one-step method for stiff systems , 1998 .

[22]  Gholamreza Hojjati,et al.  New second derivative multistep methods for stiff systems , 2006 .

[23]  M. Yousuf.  On the class of high order time stepping schemes based on Padé approximations for the numerical solution of Burgers' equation , 2008, Appl. Math. Comput..