Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs
For many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well-known backward differentiation formulae (BDF). More recently, however, new classes of formulae which can offer some important adv...
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| Vydáno v: | Journal of computational and applied mathematics Ročník 125; číslo 1; s. 117 - 130 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Elsevier B.V
15.12.2000
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | For many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well-known backward differentiation formulae (BDF). More recently, however, new classes of formulae which can offer some important advantages over BDF have emerged. In particular, some recent large-scale independent comparisons have indicated that modified extended backward differentiation formulae (MEBDF) are particularly efficient for general stiff initial value problems and for linearly implicit DAEs with index ⩽3. In the present paper we survey some of the more important theory associated with these formulae, discuss some of the practical applications where they are particularly effective, e.g., in the solution of damped highly oscillatory problems, and describe some significant recent extensions to the applicability of MEBDF codes. |
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| AbstractList | For many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well-known backward differentiation formulae (BDF). More recently, however, new classes of formulae which can offer some important advantages over BDF have emerged. In particular, some recent large-scale independent comparisons have indicated that modified extended backward differentiation formulae (MEBDF) are particularly efficient for general stiff initial value problems and for linearly implicit DAEs with index ⩽3. In the present paper we survey some of the more important theory associated with these formulae, discuss some of the practical applications where they are particularly effective, e.g., in the solution of damped highly oscillatory problems, and describe some significant recent extensions to the applicability of MEBDF codes. |
| Author | Cash, J.R. |
| Author_xml | – sequence: 1 givenname: J.R. surname: Cash fullname: Cash, J.R. email: j.cash@ic.ac.uk organization: Department of Mathematics, Imperial College of Science, South Kensington, London SW7 2BZ, UK |
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| Cites_doi | 10.1145/146847.146922 10.1007/BF01932994 10.1016/0898-1221(83)90122-0 10.1145/1218052.1218054 10.1145/235815.235818 10.6028/jres.071B.018 10.1073/pnas.38.3.235 10.1016/0898-1221(94)00092-1 10.1016/S0898-1221(98)00108-4 10.1007/BF01932291 10.1007/BF01963532 10.1016/0377-0427(85)90008-1 10.1145/356068.356073 10.1007/BF02510263 |
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| References_xml | – reference: A.C. Hindmarsh, LSODE and LSODI, two initial value ordinary differential equation solvers ACM SIGNUM Newslett. 15 (1980). – volume: 10 start-page: 58 year: 1984 end-page: 72 ident: BIB16 article-title: A performance evaluation of some FORTRAN subroutines for the solution of stiff oscillatory ordinary differential equations publication-title: ACM Trans. Math. Software – year: 1989 ident: BIB19 publication-title: The Numerical Solution of Differential-Algebraic Systems by Runge–Kutta Methods – volume: 18 start-page: 142 year: 1992 end-page: 160 ident: BIB6 article-title: An MEBDF code for stiff initial value problems publication-title: ACM Trans. Math. Software – volume: 38 start-page: 235 year: 1952 end-page: 243 ident: BIB12 article-title: Integration of stiff equations publication-title: Proc. Nat. Acad. Sci. – reference: G. 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Vander Houwen, private communication, 1999. – volume: 15 start-page: 10 year: 1975 end-page: 48 ident: BIB15 article-title: Comparing numerical methods for stiff systems of ODEs publication-title: BIT – year: 1973 ident: BIB21 publication-title: Computational Ordinary Differential Equations – year: 1997 ident: BIB2 publication-title: Solving Differential Problems by Multistep Initial and Boundary Value Methods – reference: J.C.P. Miller, Bessel Functions, Part II, Math. Tables, Vol. X, British Association for the Advancement of Sciences, CUP, 1952. – year: 1989 ident: BIB1 publication-title: Numerical Solution of Initial Value Problems in Differential-Algebraic Equations – ident: 10.1016/S0377-0427(00)00463-5_BIB27 – volume: 18 start-page: 142 year: 1992 ident: 10.1016/S0377-0427(00)00463-5_BIB6 article-title: An MEBDF code for stiff initial value problems publication-title: ACM Trans. Math. 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Appl. doi: 10.1016/0898-1221(83)90122-0 – ident: 10.1016/S0377-0427(00)00463-5_BIB20 doi: 10.1145/1218052.1218054 – year: 1996 ident: 10.1016/S0377-0427(00)00463-5_BIB18 – year: 1973 ident: 10.1016/S0377-0427(00)00463-5_BIB21 – volume: 22 start-page: 401 year: 1996 ident: 10.1016/S0377-0427(00)00463-5_BIB4 article-title: DESI methods for stiff initial value problems publication-title: ACM Trans. Math. Software doi: 10.1145/235815.235818 – volume: 71B start-page: 111 year: 1967 ident: 10.1016/S0377-0427(00)00463-5_BIB22 article-title: Numerical solution of second order linear difference equations publication-title: J. Res. Nat. Bur. Standards Math. Math. Phys. doi: 10.6028/jres.071B.018 – year: 1989 ident: 10.1016/S0377-0427(00)00463-5_BIB19 – year: 1987 ident: 10.1016/S0377-0427(00)00463-5_BIB3 – volume: 32 start-page: 497 year: 1978 ident: 10.1016/S0377-0427(00)00463-5_BIB8 article-title: An extension of Olver's method for the numerical solution of linear recurrence relations publication-title: Math. Comp. – volume: 38 start-page: 235 year: 1952 ident: 10.1016/S0377-0427(00)00463-5_BIB12 article-title: Integration of stiff equations publication-title: Proc. Nat. Acad. Sci. doi: 10.1073/pnas.38.3.235 – ident: 10.1016/S0377-0427(00)00463-5_BIB24 – ident: 10.1016/S0377-0427(00)00463-5_BIB14 – volume: 28 start-page: 45 year: 1994 ident: 10.1016/S0377-0427(00)00463-5_BIB10 article-title: Stability concepts in the numerical solution of difference and differential equations publication-title: Comput. Math. Appl. doi: 10.1016/0898-1221(94)00092-1 – volume: 36 start-page: 51 year: 1998 ident: 10.1016/S0377-0427(00)00463-5_BIB11 article-title: A comparison of some codes for the stiff oscillatory problem publication-title: Comp. Math. Appl. doi: 10.1016/S0898-1221(98)00108-4 – volume: 17 start-page: 200 year: 1977 ident: 10.1016/S0377-0427(00)00463-5_BIB25 article-title: Attainable orders of rational approximations to the exponential function with only real poles publication-title: BIT doi: 10.1007/BF01932291 – volume: 3 start-page: 27 year: 1963 ident: 10.1016/S0377-0427(00)00463-5_BIB13 article-title: A special stability problem for linear multistep methods publication-title: BIT doi: 10.1007/BF01963532 – volume: 12 start-page: 77 year: 1985 ident: 10.1016/S0377-0427(00)00463-5_BIB17 article-title: Automatic integration of Euler–Lagrange equations with constraints publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(85)90008-1 – volume: 10 start-page: 58 year: 1984 ident: 10.1016/S0377-0427(00)00463-5_BIB16 article-title: A performance evaluation of some FORTRAN subroutines for the solution of stiff oscillatory ordinary differential equations publication-title: ACM Trans. Math. Software doi: 10.1145/356068.356073 – volume: 38 start-page: 612 year: 1998 ident: 10.1016/S0377-0427(00)00463-5_BIB26 article-title: A stability result for general linear methods with characteristic function having real poles only publication-title: BIT doi: 10.1007/BF02510263 |
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| Title | Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs |
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