Optimal extended one-step schemes of exponential type for stiff initial-value problems

References

• Keller HB (1968) Numerical methods for two-point boundary value problems, Blaisdell Publishing Company London
   Google Scholar
• Carroll J (1991) Sufficient conditions for uniformly second-order convergent schemes for stiff initial-value problems In: R. Vichnevetsky, J. H. H. Miller (Eds.) Proc. of the 13th IMACS World Congress on Computation and Applied Mathematics, Trinity College Dublin 285
   Google Scholar
• Carroll , J . (1993) . A matrical exponentially fitted scheme for the numerical solution of stiff initial value problems . Comput. Math. Appl. , 26 : 57 – 64 .
   Google Scholar
• Doolan EP Miller JJH Schilders WHA (1980) Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press Dublin
   Google Scholar
• Miller , JJH . (1986) . Optimal uniform difference schemes for linear initial-value problems . Comput. Math. Appl. , 12 : 1209 – 1215 .
   Google Scholar
• Salama , AA . (1996) . A class of exponential methods for stiff initial-value problems . Int. J. Comp. Math. , 62 : 183 – 198 .
   Web of Science ®Google Scholar
• Hairer E Wanner G (1996) Solving Ordinary Differential Equations II 2nd ed. Springer-Verlag Berlin
   Google Scholar
• Hochbruck , M , Lubich , C and Selhofer , H . (1998) . Exponential integrators for large systems of differential equations . SIAM J. Sci. Comput. , 19 : 1552 – 1574 .
   Web of Science ®Google Scholar
• O'Reilly , MJ . (1987) . A uniform scheme for the singularly perturbed Riccati equations . Numer. Math. , 50 : 483 – 501 .
   Google Scholar
• Salama , AA . (2000) . Higher order and optimal schemes for stiff initial-value problems . Math. Japonica , 52 : 377 – 386 .
   Google Scholar
• Usmani , RA and Agarwal , RP . (1985) . An A-stable extended trapezoidal rule for the numerical integration of ordinary differential equations . Comput. Math. Appl. , 11 : 1183 – 1191 .
   Web of Science ®Google Scholar
• Jacques , IB . (1989) . Extended one-step methods for the numerical solution of ordinary differential equations . Int. J. Comput. Math. , 29 : 247 – 255 .
   Web of Science ®Google Scholar
• Kondrat , DM and Jacques , IB . (1992) . Extended A-stable two-step methods for the numerical solution of ordinary differential equations . Int. J. Comput. Math. , 42 : 117 – 124 .
   Web of Science ®Google Scholar
• Chawla , MM , Al-Zanaidi , MA and Al-Sahhar , MS . (1994) . Stabilized fourth order extended methods for the numerical solution of ODEs . Int. J. Comput. Math. , 52 : 99 – 107 .
   Web of Science ®Google Scholar
• Chawla , MM , Al-Zanaidi , MA and Al-Sahhar , MS . (1995) . A class of stabilized extended one-step methods for the numerical solution of ODEs . Comput. Math. Appl. , 29 : 79 – 84 .
   Web of Science ®Google Scholar
• Chawla , MM , Zanaidi , MA and Evans , DJ . (1996) . A class of generalized trapezoidal formulas for the numerical integration of y′ = f(x, y) . Int. J. Comput. Math. , 62 : 131 – 142 .
   Web of Science ®Google Scholar
• De Meyer , H , Van Hecke , T , Vanden Berghe , G and Van Daele , M . (1996) . Multistride L-Stable fourth-order methods for the numerical solution of ODEs . Comput. Math. Appl. , 32 : 1 – 8 .
   Google Scholar
• Carroll , J . (1992) . Sufficient conditions for uniformly second-order convergent schemes for stiff initial-value problems . Comput. Math. Appl. , 24 : 105 – 116 .
   Google Scholar
• Gourdon X Sebah P (1999) Numbers, Constants and Computation World Wide Web site at the address:, http://numbers.computation.free.fr/Constants/constants.html
   Google Scholar
• Gourdon X Sebah P (2002) Introduction on Bernoulli's numbers World Wide Web site at the address:, http://numbers.computation.free.fr/Constants/constants.html
   Google Scholar
• Farrell , PA . (1987) . Uniform and optimal schemes for stiff initial-value problems . Comput. Math. Appl. , 13 : 925 – 936 .
   Web of Science ®Google Scholar
• Van Hecke , T , Vanden Berghe , G , Van Daele , M and De Meyer , H . (1997) . A variable-step variable-order algorithm for systems of stiff ODEs . Int. J. Comput. Math. , 63 : 149 – 157 .
   Web of Science ®Google Scholar
• Doolan EP Schilders WHA (1980) Uniformly convergent methods for stiff initial value problems In: J. J. H. Miller (Ed.) Boundary and Interior Layers-Computational and Asymptotic Methods, Boole Press Dublin pp. 256–259
   Google Scholar
• Frank , R and Ueberhuber , CW . (1977) . Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations . BIT , 17 : 146 – 159 .
   Google Scholar
• Van Niekerk , FD . (1988) . Rational one-step methods for initial value problems . Comput. Math. Appl. , 16 : 1035 – 1039 .
   Web of Science ®Google Scholar
• De Meyer , H , Vanden Berghe , G and Vanthournout , J . (1991) . Modified backward differentiation methods of the Adams-type based on exponential interpolation . Comput. Math. Appl. , 21 : 171 – 179 .
   Google Scholar
• Vanthournout , J , Vanden Berghe , G and De Meyer , H . (1990) . Families of backward differentiation methods based on a new type of mixed interpolation . Comput. Math. Appl. , 20 : 19 – 30 .
   Web of Science ®Google Scholar
• Finlayson BA (1980) Nonlinear Analysis in Chemical Engineering, McGraw-Hill New York
   Google Scholar