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Volumn 25, Issue 1, 2005, Pages 29-49

Construction and implementation of general linear methods for ordinary differential equations: A review

Author keywords

General linear methods; Implementation aspects; Local error estimation; Order and stage order; Runge Kutta stability

Indexed keywords

DIFFERENTIAL EQUATIONS; ERROR ANALYSIS; INTEGRATION; NONLINEAR EQUATIONS; RUNGE KUTTA METHODS;

EID: 28844463400     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-004-4631-9     Document Type: Conference Paper
Times cited : (8)

References (46)
  • 3
    • 38249003919 scopus 로고
    • Diagonally-implicit multi-stage integration methods
    • Butcher, J. C. (1993). Diagonally-implicit multi-stage integration methods. Appl. Numer. Math. 11, 347-363.
    • (1993) Appl. Numer. Math. , vol.11 , pp. 347-363
    • Butcher, J.C.1
  • 4
    • 0039344085 scopus 로고
    • A transformation for the analysis of DIMSIMs
    • Butcher, J. C. (1994). A transformation for the analysis of DIMSIMs. BIT 34, 25-32.
    • (1994) BIT , vol.34 , pp. 25-32
    • Butcher, J.C.1
  • 5
    • 0012458515 scopus 로고    scopus 로고
    • General linear methods for stiff differential equations
    • Butcher, J. C. (2001). General linear methods for stiff differential equations. BIT 41, 193-221.
    • (2001) BIT , vol.41 , pp. 193-221
    • Butcher, J.C.1
  • 7
    • 0042214275 scopus 로고    scopus 로고
    • Experiments with a variable-order type J. DIMSIM code
    • Butcher, J. C., Chartier, P., and Jackiewicz, Z. (1999). Experiments with a variable-order type J. DIMSIM code. Numer. Algorithms 22, 237-261.
    • (1999) Numer. Algorithms , vol.22 , pp. 237-261
    • Butcher, J.C.1    Chartier, P.2    Jackiewicz, Z.3
  • 8
    • 0001074434 scopus 로고
    • Diagonally implicit general linear methods for ordinary differential equations
    • Butcher, J. C., and Jackiewicz, Z. (1993). Diagonally implicit general linear methods for ordinary differential equations. BIT 33, 452-412.
    • (1993) BIT , vol.33 , pp. 452-1412
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 9
    • 0030263281 scopus 로고    scopus 로고
    • Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations
    • Butcher, J. C., and Jackiewicz, Z. (1996). Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations. Appl. Numer. Math. 21, 385-415.
    • (1996) Appl. Numer. Math. , vol.21 , pp. 385-415
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 10
    • 0000452884 scopus 로고    scopus 로고
    • Implementation of diagonally implicit multistage integration methods for ordinary differential equations
    • Butcher, J. C, and Jackiewicz, Z. (1997). Implementation of diagonally implicit multistage integration methods for ordinary differential equations. SIAM J. Numer. Anal. 34, 2119-2141.
    • (1997) SIAM J. Numer. Anal. , vol.34 , pp. 2119-2141
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 11
    • 0032068326 scopus 로고    scopus 로고
    • Construction of high order diagonally implicit multistage integration methods for ordinary differential equations
    • Butcher, J. C., and Jackiewicz, Z. (1998). Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Appl. Numer. Math. 27, 1-12.
    • (1998) Appl. Numer. Math. , vol.27 , pp. 1-12
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 12
    • 0042318570 scopus 로고    scopus 로고
    • A reliable error estimation for DIMSIMs
    • Butcher, J. C., and Jackiewicz, Z. (2001). A reliable error estimation for DIMSIMs. BIT 41, 656-665.
    • (2001) BIT , vol.41 , pp. 656-665
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 13
    • 0036441464 scopus 로고    scopus 로고
    • Error estimation for Nordsieck methods
    • Butcher, J. C., and Jackiewicz, Z. (2002). Error estimation for Nordsieck methods. Numer. Algorithms 31, 75-85.
    • (2002) Numer. Algorithms , vol.31 , pp. 75-85
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 14
    • 0345060035 scopus 로고    scopus 로고
    • A new approach to error estimation for general linear methods
    • Butcher, J. C., and Jackiewicz, Z. (2003). A new approach to error estimation for general linear methods. Numer. Math. 95, 487-502.
    • (2003) Numer. Math. , vol.95 , pp. 487-502
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 15
    • 4043085577 scopus 로고    scopus 로고
    • Construction of general linear methods with Runge-Kutta stability properties
    • Butcher, J. C., and Jackiewicz, Z. (2004) Construction of general linear methods with Runge-Kutta stability properties. Numer. Algorithms 36, 53-72.
    • (2004) Numer. Algorithms , vol.36 , pp. 53-72
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 16
    • 8144230572 scopus 로고    scopus 로고
    • Unconditionally stable general linear methods for ordinary differential equations
    • Butcher, J. C., and Jackiewicz, Z. (2004) Unconditionally stable general linear methods for ordinary differential equations BIT. 44, 557-570.
    • (2004) BIT. , vol.44 , pp. 557-570
    • Butcher, J.C.1    Jackiewicz, Z.2
  • 17
    • 0031559380 scopus 로고    scopus 로고
    • A nonlinear optimization approach to the construction of general linear methods of high order
    • Butcher, J. C., Jackiewicz, Z., and Mittelmann, H. D. (1997). A nonlinear optimization approach to the construction of general linear methods of high order. J. Comp. Appl. Math. 81, 181-196.
    • (1997) J. Comp. Appl. Math. , vol.81 , pp. 181-196
    • Butcher, J.C.1    Jackiewicz, Z.2    Mittelmann, H.D.3
  • 19
    • 4043179733 scopus 로고    scopus 로고
    • The construction of practical general linear methods
    • Butcher, J. C., and Wright, W. M.AM (2003) The construction of practical general linear methods BIT. 43, 695-721.
    • (2003) BIT. , vol.43 , pp. 695-721
    • Butcher, J.C.1    Wright, W.M.2
  • 20
    • 0041720995 scopus 로고    scopus 로고
    • The potential of parallel multi-value methods for the simulation of large real-life problems
    • Chartier, P. (1998). The potential of parallel multi-value methods for the simulation of large real-life problems. CWI Quarterly 11, 7-32.
    • (1998) CWI Quarterly , vol.11 , pp. 7-32
    • Chartier, P.1
  • 21
    • 84976747891 scopus 로고
    • Algorithm 573, NL2SOL-An adaptive nonlinear least-squares algorithm
    • Dennis, J. E., Gay, D. M., and Welsch, R. E. (1981). Algorithm 573, NL2SOL-An adaptive nonlinear least-squares algorithm. ACM Trans. Math. Software 7, 369-383.
    • (1981) ACM Trans. Math. Software , vol.7 , pp. 369-383
    • Dennis, J.E.1    Gay, D.M.2    Welsch, R.E.3
  • 23
    • 0023401540 scopus 로고
    • Efficient numerical path following beyond critical points
    • Deuflhard, P., Fiedler, B., and Kunkel, P. (1987). Efficient numerical path following beyond critical points. SIAM J. Numer. Anal. 24, 912-927.
    • (1987) SIAM J. Numer. Anal. , vol.24 , pp. 912-927
    • Deuflhard, P.1    Fiedler, B.2    Kunkel, P.3
  • 24
    • 0023349566 scopus 로고
    • Automatic selection of initial stepsize for an ODE solver
    • Gladwell, I., Shampine, L. E, and Brankin, R. W (1987). Automatic selection of initial stepsize for an ODE solver. J. Comput. Appl. Math. 18, 175-192.
    • (1987) J. Comput. Appl. Math. , vol.18 , pp. 175-192
    • Gladwell, I.1    Shampine, L.E.2    Brankin, R.W.3
  • 27
    • 28844446704 scopus 로고    scopus 로고
    • Step-control stability for diagonally implicit multistage integration methods
    • Jackiewicz, Z. (2000). Step-control stability for diagonally implicit multistage integration methods. New Zealand J. of Math. 29, 193-201.
    • (2000) New Zealand J. of Math. , vol.29 , pp. 193-201
    • Jackiewicz, Z.1
  • 28
    • 0036679991 scopus 로고    scopus 로고
    • Implementation of DIMSIMs for stiff differential systems
    • Jackiewicz, Z. (2002). Implementation of DIMSIMs for stiff differential systems. Appl. Numer. Math. 42, 251-267.
    • (2002) Appl. Numer. Math. , vol.42 , pp. 251-267
    • Jackiewicz, Z.1
  • 29
    • 0032671026 scopus 로고    scopus 로고
    • Exploiting structure in the construction of DIMSIMs
    • Jackiewicz, Z., and Mittelmann, H. D. (1999). Exploiting structure in the construction of DIMSIMs. J. Comp. Appl. Math. 107, 233-239.
    • (1999) J. Comp. Appl. Math. , vol.107 , pp. 233-239
    • Jackiewicz, Z.1    Mittelmann, H.D.2
  • 30
    • 0001472260 scopus 로고    scopus 로고
    • General linear methods with external stages of different orders
    • Jackiewicz, Z., and Vermiglio, R. (1996). General linear methods with external stages of different orders. BIT 36, 688-712.
    • (1996) BIT , vol.36 , pp. 688-712
    • Jackiewicz, Z.1    Vermiglio, R.2
  • 31
    • 0007082106 scopus 로고
    • Variable stepsize diagonally implicit multistage integration methods for ordinary differential equations
    • Jackiewicz, Z., Vermiglio, R., and Zennaro, M. (1995). Variable stepsize diagonally implicit multistage integration methods for ordinary differential equations. Appl. Numer. Math. 16, 343-367.
    • (1995) Appl. Numer. Math. , vol.16 , pp. 343-367
    • Jackiewicz, Z.1    Vermiglio, R.2    Zennaro, M.3
  • 33
    • 84976786620 scopus 로고
    • Algorithm 596: A program for a locally-parametrized continuation process
    • Rheinboldt, W. C., and Burkardt, J. V. (1983). Algorithm 596: a program for a locally-parametrized continuation process. ACM Trans. Math. Software 9, 236-241.
    • (1983) ACM Trans. Math. Software , vol.9 , pp. 236-241
    • Rheinboldt, W.C.1    Burkardt, J.V.2
  • 34
    • 0001014282 scopus 로고
    • Implementation of implicit formulas for the solution of ODEs
    • Shampine, L. F. (1980). Implementation of implicit formulas for the solution of ODEs. SIAM J. Sci. Stat. Comput. 1, 103-118.
    • (1980) SIAM J. Sci. Stat. Comput. , vol.1 , pp. 103-118
    • Shampine, L.F.1
  • 36
    • 0021505022 scopus 로고
    • Error estimators for stiff differential equations
    • Shampine, L. F., and Baca, L. S. (1984). Error estimators for stiff differential equations. J. Comput. Appl. Math. 11, 197-207.
    • (1984) J. Comput. Appl. Math. , vol.11 , pp. 197-207
    • Shampine, L.F.1    Baca, L.S.2
  • 38
    • 0002396750 scopus 로고    scopus 로고
    • The automatic control in numerical integration
    • Söderlind, G. (1998). The automatic control in numerical integration. CWI Quarterly 11, 55-74.
    • (1998) CWI Quarterly , vol.11 , pp. 55-74
    • Söderlind, G.1
  • 39
    • 0036441379 scopus 로고    scopus 로고
    • Automatic control and adaptive time-stepping
    • Söderlind, G. (2002). Automatic control and adaptive time-stepping. Numer. Algorithms 31, 281-310.
    • (2002) Numer. Algorithms , vol.31 , pp. 281-310
    • Söderlind, G.1
  • 40
    • 3042616322 scopus 로고    scopus 로고
    • Digital filters in adaptive time-stepping
    • Söderlind, G. (2003). Digital filters in adaptive time-stepping. ACM Trans. Math. Software 29, 1-26.
    • (2003) ACM Trans. Math. Software , vol.29 , pp. 1-26
    • Söderlind, G.1
  • 43
    • 0023415939 scopus 로고
    • HOMPACK: A suite od codes for globally convergent homotopy algorithms
    • Watson, L. T., Billups, S. C., and Morgan, A. P. (1987). HOMPACK: a suite od codes for globally convergent homotopy algorithms. ACM Trans. Math. Software 13, 281-310.
    • (1987) ACM Trans. Math. Software , vol.13 , pp. 281-310
    • Watson, L.T.1    Billups, S.C.2    Morgan, A.P.3
  • 44
    • 0035616409 scopus 로고    scopus 로고
    • The construction of an order 4 DIMSIMs for ordinary differential equations
    • Wright, W. M. (2001). The construction of an order 4 DIMSIMs for ordinary differential equations. Numer. Algorithms 26, 123-130.
    • (2001) Numer. Algorithms , vol.26 , pp. 123-130
    • Wright, W.M.1
  • 46
    • 28844435471 scopus 로고
    • 3rd ed., AT & T Lab-oratories, Murray Hill, NJ. The public part of this library is available at
    • The PORT Mathematical Subroutine Library. (1984). 3rd ed., AT & T Lab-oratories, Murray Hill, NJ. The public part of this library is available at http://www.netlib.org/port/.
    • (1984)


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