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Volumn 22, Issue 1-3 SPEC. ISS., 1996, Pages 225-236

Runge-Kutta research at Toronto

Author keywords

Initial value problems; Ordinary differential equations; Runge Kutta methods

Indexed keywords

ALGEBRA; COMPUTATIONAL METHODS; COMPUTER SOFTWARE; DIFFERENTIAL EQUATIONS; ERROR CORRECTION; PROBLEM SOLVING;

EID: 0030287582     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0168-9274(96)00034-7     Document Type: Article
Times cited : (9)

References (85)
  • 3
    • 27844591085 scopus 로고
    • Numerical treatment of ordinary differential equations by extrapolation methods
    • R. Bulirsch and J. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods, Numer. Math. 8 (1966) 1-13.
    • (1966) Numer. Math. , vol.8 , pp. 1-13
    • Bulirsch, R.1    Stoer, J.2
  • 5
    • 0001651643 scopus 로고
    • Nonlinearly preconditioned Krylov subspace methods for discrete-Newton algorithms
    • T.F. Chan and K.R. Jackson, Nonlinearly preconditioned Krylov subspace methods for discrete-Newton algorithms, SIAM J. Sci. Statist. Comput. 5 (1984) 533-542.
    • (1984) SIAM J. Sci. Statist. Comput. , vol.5 , pp. 533-542
    • Chan, T.F.1    Jackson, K.R.2
  • 6
    • 0001503361 scopus 로고
    • The use of iterative linear-equation solvers in codes for large systems of stiff IVPs for ODEs
    • T.F. Chan and K.R. Jackson, The use of iterative linear-equation solvers in codes for large systems of stiff IVPs for ODEs, SIAM J. Sci. Statist. Comput. 1 (1986) 378-417.
    • (1986) SIAM J. Sci. Statist. Comput. , vol.1 , pp. 378-417
    • Chan, T.F.1    Jackson, K.R.2
  • 7
    • 0347742477 scopus 로고
    • Alternating-direction incomplete factorizations
    • T.F. Chan, K.R. Jackson and B. Zhu, Alternating-direction incomplete factorizations, SIAM J. Numer. Anal. 20 (1983) 239-257.
    • (1983) SIAM J. Numer. Anal. , vol.20 , pp. 239-257
    • Chan, T.F.1    Jackson, K.R.2    Zhu, B.3
  • 8
    • 0011566457 scopus 로고
    • Convergence and stability in the numerical integration of ordinary differential equations
    • G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956) 33-53.
    • (1956) Math. Scand. , vol.4 , pp. 33-53
    • Dahlquist, G.1
  • 9
    • 84975603488 scopus 로고
    • Nonlinear coupled mode equations on a finite interval: A numerical procedure
    • Special issue on Numerical Simulation in Guided-Wave Optics and Electroptics.
    • C.M. de Sterke, K.R. Jackson and B.D. Robert, Nonlinear coupled mode equations on a finite interval: a numerical procedure, J. Opt. Soc. Amer. B 8 (1991) 403-412 (Special issue on Numerical Simulation in Guided-Wave Optics and Electroptics).
    • (1991) J. Opt. Soc. Amer. B , vol.8 , pp. 403-412
    • De Sterke, C.M.1    Jackson, K.R.2    Robert, B.D.3
  • 14
    • 85086526260 scopus 로고    scopus 로고
    • DIMSEMs - Diagonally implicit single-eigenvalue methods for the numerical solution of stiff ODEs on parallel computers
    • to be submitted
    • R.F. Enenkel and K.R. Jackson, DIMSEMs - diagonally implicit single-eigenvalue methods for the numerical solution of stiff ODEs on parallel computers, Adv. Comput. Math., to be submitted.
    • Adv. Comput. Math.
    • Enenkel, R.F.1    Jackson, K.R.2
  • 15
    • 0010965403 scopus 로고
    • Analysis of error control strategies for continuous Runge-Kutta methods
    • W.H. Enright, Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal. 26 (1989) 588-599.
    • (1989) SIAM J. Numer. Anal. , vol.26 , pp. 588-599
    • Enright, W.H.1
  • 16
    • 0001743435 scopus 로고
    • A new error-control for initial value solvers
    • W.H. Enright, A new error-control for initial value solvers, Appl. Math. Comput. 31 (1989) 288-301.
    • (1989) Appl. Math. Comput. , vol.31 , pp. 288-301
    • Enright, W.H.1
  • 17
    • 0027684529 scopus 로고
    • The relative efficiency of alternative defect control schemes for high order Runge-Kutta formulas
    • W.H. Enright, The relative efficiency of alternative defect control schemes for high order Runge-Kutta formulas, SIAM J. Numer. Anal. 30 (1993) 1419-1445.
    • (1993) SIAM J. Numer. Anal. , vol.30 , pp. 1419-1445
    • Enright, W.H.1
  • 19
    • 0347742479 scopus 로고    scopus 로고
    • A delay differential equation solver based on a continuous Runge-Kutta method
    • submitted
    • W.H. Enright and H. Hayashi, A delay differential equation solver based on a continuous Runge-Kutta method, SIAM J. Sci. Comput., submitted.
    • SIAM J. Sci. Comput.
    • Enright, W.H.1    Hayashi, H.2
  • 20
    • 0040278315 scopus 로고
    • Parallel defect control
    • W.H. Enright and D.J. Higham, Parallel defect control, BIT 31 (1991) 647-663.
    • (1991) BIT , vol.31 , pp. 647-663
    • Enright, W.H.1    Higham, D.J.2
  • 21
    • 0004073948 scopus 로고
    • A survey of the explicit Runge-Kutta method
    • Department of Computer Science, University of Toronto
    • W.H. Enright, D.J. Higham, B. Owren and P.W. Sharp, A survey of the explicit Runge-Kutta method, Technical Report No. 291, Department of Computer Science, University of Toronto (1994).
    • (1994) Technical Report No. 291 , vol.291
    • Enright, W.H.1    Higham, D.J.2    Owren, B.3    Sharp, P.W.4
  • 22
    • 0029509590 scopus 로고
    • Interpolating Runge-Kutta methods for vanishing delay differential equations
    • W.H. Enright and M. Hu, Interpolating Runge-Kutta methods for vanishing delay differential equations, Computing 55 (1995) 223-236.
    • (1995) Computing , vol.55 , pp. 223-236
    • Enright, W.H.1    Hu, M.2
  • 23
    • 0345850952 scopus 로고
    • Test results on initial value methods for non-stiff ordinary differential equations
    • W.H. Enright and T.E. Hull, Test results on initial value methods for non-stiff ordinary differential equations, SIAM J. Numer. Anal. 13 (1976) 944-961.
    • (1976) SIAM J. Numer. Anal. , vol.13 , pp. 944-961
    • Enright, W.H.1    Hull, T.E.2
  • 25
    • 0000207879 scopus 로고
    • Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants
    • W.H. Enright, K.R. Jackson, S.P. Nørsett and P.G. Thomsen, Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants, Appl. Math. Comput. 27 (1988) 313-335.
    • (1988) Appl. Math. Comput. , vol.27 , pp. 313-335
    • Enright, W.H.1    Jackson, K.R.2    Nørsett, S.P.3    Thomsen, P.G.4
  • 26
    • 0042919986 scopus 로고
    • The practical implications of order reduction for implicit Runge-Kutta methods
    • W.H. Enright and C. MacDonald, The practical implications of order reduction for implicit Runge-Kutta methods, Numer. Algorithms 2 (1992) 351-370.
    • (1992) Numer. Algorithms , vol.2 , pp. 351-370
    • Enright, W.H.1    MacDonald, C.2
  • 27
    • 0022889135 scopus 로고
    • Efficient classes of Runge-Kutta methods for two point boundary value problems
    • W.H. Enright and P. Muir, Efficient classes of Runge-Kutta methods for two point boundary value problems, Computing 37 (1986) 315-334.
    • (1986) Computing , vol.37 , pp. 315-334
    • Enright, W.H.1    Muir, P.2
  • 28
    • 2142782818 scopus 로고    scopus 로고
    • A Runge-Kutta type boundary value ODE solver with defect control
    • W.H. Enright and P. Muir, A Runge-Kutta type boundary value ODE solver with defect control, SIAM J. Sci. Comput. 17 (1996) 479-497.
    • (1996) SIAM J. Sci. Comput. , vol.17 , pp. 479-497
    • Enright, W.H.1    Muir, P.2
  • 29
    • 0342590227 scopus 로고
    • Two Fortran packages for assessing initial value methods
    • Department of Computer Science, University of Toronto
    • W.H. Enright and J.D. Pryce, Two Fortran packages for assessing initial value methods, Technical Report No. 167, Department of Computer Science, University of Toronto (1983).
    • (1983) Technical Report No. 167 , vol.167
    • Enright, W.H.1    Pryce, J.D.2
  • 30
    • 0023312263 scopus 로고
    • Two FORTRAN packages for assessing initial value methods
    • W.H. Enright and J.D. Pryce, Two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Software 13 (1987) 1-27.
    • (1987) ACM Trans. Math. Software , vol.13 , pp. 1-27
    • Enright, W.H.1    Pryce, J.D.2
  • 31
    • 0005281788 scopus 로고
    • Detecting and locating a singular point in the numerical solution of IVPs for ODEs
    • W.H. Enright and H. Suhartanto, Detecting and locating a singular point in the numerical solution of IVPs for ODEs, Computing 48 (1992) 161-175.
    • (1992) Computing , vol.48 , pp. 161-175
    • Enright, W.H.1    Suhartanto, H.2
  • 33
    • 0023511191 scopus 로고
    • Interpolants for Runge-Kutta-Nyström methods
    • J.M. Fine, Interpolants for Runge-Kutta-Nyström methods, Computing 39 (1987) 27-42.
    • (1987) Computing , vol.39 , pp. 27-42
    • Fine, J.M.1
  • 34
    • 0041590181 scopus 로고
    • DETEST: A program for comparing numerical methods for ordinary differential equations
    • Department of Computer Science, University of Toronto
    • G. Hall, W.H. Enright, T.E. Hull and A.E. Sedgwick, DETEST: A program for comparing numerical methods for ordinary differential equations, Technical Report No. 60, Department of Computer Science, University of Toronto (1973).
    • (1973) Technical Report No. 60 , vol.60
    • Hall, G.1    Enright, W.H.2    Hull, T.E.3    Sedgwick, A.E.4
  • 38
    • 0012211602 scopus 로고
    • Global error versus tolerance for explicit Runge-Kutta methods
    • D.J. Higham, Global error versus tolerance for explicit Runge-Kutta methods, IMA J. Numer. Anal. 11 (1991) 457-480.
    • (1991) IMA J. Numer. Anal. , vol.11 , pp. 457-480
    • Higham, D.J.1
  • 39
    • 0343359688 scopus 로고
    • Runge-Kutta defect control using Hermite-Birkoff interpolation
    • D.J. Higham, Runge-Kutta defect control using Hermite-Birkoff interpolation, SIAM J. Sci. Statist. Comput. 12 (1991) 991-999.
    • (1991) SIAM J. Sci. Statist. Comput. , vol.12 , pp. 991-999
    • Higham, D.J.1
  • 40
    • 0346481900 scopus 로고
    • A search for optimum methods for the numerical integration of ordinary differential equations
    • T.E. Hull, A search for optimum methods for the numerical integration of ordinary differential equations, SIAM Rev. 9 (1967) 647-654.
    • (1967) SIAM Rev. , vol.9 , pp. 647-654
    • Hull, T.E.1
  • 41
    • 84976765421 scopus 로고
    • The effectiveness of numerical methods for ordinary differential equations
    • T.E. Hull, The effectiveness of numerical methods for ordinary differential equations, SIAM Stud. Numer. Anal. 2 (1968) 114-121.
    • (1968) SIAM Stud. Numer. Anal. , vol.2 , pp. 114-121
    • Hull, T.E.1
  • 42
    • 0346481895 scopus 로고
    • The numerical integration of ordinary differential equations
    • North-Holland, Amsterdam
    • T.E. Hull, The numerical integration of ordinary differential equations, in: Proceedings IFIP Congress 68 (North-Holland, Amsterdam, 1969) 39-53.
    • (1969) Proceedings IFIP Congress 68 , pp. 39-53
    • Hull, T.E.1
  • 45
    • 0012087157 scopus 로고
    • Correctness of numerical software
    • L.D. Fosdick. ed., North-Holland, Amsterdam
    • T.E. Hull, Correctness of numerical software, in: L.D. Fosdick. ed., Performance of Evaluation of Numerical Software (North-Holland, Amsterdam, 1979) 3-15.
    • (1979) Performance of Evaluation of Numerical Software , pp. 3-15
    • Hull, T.E.1
  • 46
    • 0347111994 scopus 로고
    • A structure for programs that solve ordinary differential equations
    • Department of Computer Science, University of Toronto
    • T.E. Hull and W.H. Enright, A structure for programs that solve ordinary differential equations, Technical Report No. 66, Department of Computer Science, University of Toronto (1974).
    • (1974) Technical Report No. 66 , vol.66
    • Hull, T.E.1    Enright, W.H.2
  • 47
    • 0015484437 scopus 로고
    • Comparing numerical methods for ordinary differential equations
    • see also Errata: SIAM J. Numer. Anal. 11 (1974) 681
    • T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9 (1972) 603-637; see also Errata: SIAM J. Numer. Anal. 11 (1974) 681.
    • (1972) SIAM J. Numer. Anal. , vol.9 , pp. 603-637
    • Hull, T.E.1    Enright, W.H.2    Fellen, B.M.3    Sedgwick, A.E.4
  • 48
    • 0003413506 scopus 로고
    • User's guide for DVERK - A subroutine for solving non-stiff ODEs
    • Department of Computer Science, University of Toronto
    • T.E. Hull, W.H. Enright and K.R. Jackson, User's guide for DVERK - a subroutine for solving non-stiff ODEs, Technical Report No. 100, Department of Computer Science, University of Toronto (1976).
    • (1976) Technical Report No. 100 , vol.100
    • Hull, T.E.1    Enright, W.H.2    Jackson, K.R.3
  • 50
    • 84968514040 scopus 로고
    • Optimum Runge-Kutta methods
    • T.E. Hull and R.L. Johnston, Optimum Runge-Kutta methods, Math. Comp. 18 (1964) 306-310.
    • (1964) Math. Comp. , vol.18 , pp. 306-310
    • Hull, T.E.1    Johnston, R.L.2
  • 53
    • 0026226272 scopus 로고
    • A survey of parallel numerical methods for initial value problems for ordinary differential equations
    • K.R. Jackson, A survey of parallel numerical methods for initial value problems for ordinary differential equations, IEEE Trans. Magnetics 27 (1991) 3792-3797.
    • (1991) IEEE Trans. Magnetics , vol.27 , pp. 3792-3797
    • Jackson, K.R.1
  • 54
    • 0347742469 scopus 로고
    • An application of Runge-Kutta predictor-corrector methods to two system of hyperbolic equations arising in optics
    • W.F. Ames, ed., Atlanta, GA
    • K.R. Jackson, An application of Runge-Kutta predictor-corrector methods to two system of hyperbolic equations arising in optics, in: W.F. Ames, ed., Fourteenth IMACS World Congress, Atlanta, GA (1994) 1329-1331.
    • (1994) Fourteenth IMACS World Congress , pp. 1329-1331
    • Jackson, K.R.1
  • 55
    • 0030081652 scopus 로고    scopus 로고
    • The numerical solution of large systems of stiff IVPs for ODEs
    • K.R. Jackson, The numerical solution of large systems of stiff IVPs for ODEs, Appl. Numer. Math. 20 (1996) 1-16.
    • (1996) Appl. Numer. Math. , vol.20 , pp. 1-16
    • Jackson, K.R.1
  • 56
    • 0345850948 scopus 로고
    • A theoretical criterion for comparing Runge-Kutta formulas
    • K.R. Jackson, W.H. Enright and T.E. Hull, A theoretical criterion for comparing Runge-Kutta formulas, SIAM J. Numer. Anal. 15 (1978) 618-641.
    • (1978) SIAM J. Numer. Anal. , vol.15 , pp. 618-641
    • Jackson, K.R.1    Enright, W.H.2    Hull, T.E.3
  • 57
    • 0347742471 scopus 로고    scopus 로고
    • An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values
    • to appear
    • K.R. Jackson, A. Kværnø and S.P. Nørsett, An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values, BIT. to appear.
    • BIT
    • Jackson, K.R.1    Kværnø, A.2    Nørsett, S.P.3
  • 58
    • 0028524312 scopus 로고
    • The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas
    • K.R. Jackson, A. Kværnø and S.P. Nørsett, The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas, Appl. Numer. Math. 15 (1994) 341-356.
    • (1994) Appl. Numer. Math. , vol.15 , pp. 341-356
    • Jackson, K.R.1    Kværnø, A.2    Nørsett, S.P.3
  • 59
    • 0001968970 scopus 로고
    • The potential for parallelism in Runge-Kutta methods, Part 1: RK formulas in standard form
    • K.R. Jackson and S.P. Nørsett, The potential for parallelism in Runge-Kutta methods, Part 1: RK formulas in standard form, SIAM J. Numer. Anal. 32 (1995) 49-82.
    • (1995) SIAM J. Numer. Anal. , vol.32 , pp. 49-82
    • Jackson, K.R.1    Nørsett, S.P.2
  • 60
    • 0038048652 scopus 로고
    • The parallel solution of ABD systems arising in numerical methods for BVPs for ODEs
    • Department of Computer Science, University of Toronto
    • K.R. Jackson and R.N. Pancer, The parallel solution of ABD systems arising in numerical methods for BVPs for ODEs, Technical Report No. 255, Department of Computer Science, University of Toronto (1992).
    • (1992) Technical Report No. 255 , vol.255
    • Jackson, K.R.1    Pancer, R.N.2
  • 61
    • 0040280167 scopus 로고
    • Adaptive linear equation solvers in codes for large stiff systems of ODEs
    • K.R. Jackson and W.L. Seward, Adaptive linear equation solvers in codes for large stiff systems of ODEs, SIAM J. Sci. Comput. 14 (1993) 800-823.
    • (1993) SIAM J. Sci. Comput. , vol.14 , pp. 800-823
    • Jackson, K.R.1    Seward, W.L.2
  • 63
    • 0005698025 scopus 로고
    • A computable error-bound for systems of ordinary differential equations
    • W.M. Kahan, A computable error-bound for systems of ordinary differential equations, Abstract SIAM Rev. 8 (1966) 568-569.
    • (1966) SIAM Rev. , vol.8 , pp. 568-569
    • Kahan, W.M.1
  • 66
    • 21344485778 scopus 로고
    • Order barriers and characterizations of continuous mono-implicit Runge-Kutta schemes
    • P. Muir and B. Owren, Order barriers and characterizations of continuous mono-implicit Runge-Kutta schemes, Math. Comp. 61 (1993) 675-699.
    • (1993) Math. Comp. , vol.61 , pp. 675-699
    • Muir, P.1    Owren, B.2
  • 69
    • 84966199026 scopus 로고
    • Order barriers for continuous explicit Runge-Kutta methods
    • B. Owren and M. Zennaro. Order barriers for continuous explicit Runge-Kutta methods, Math. Comp. 56 (1991) 645-661.
    • (1991) Math. Comp. , vol.56 , pp. 645-661
    • Owren, B.1    Zennaro, M.2
  • 70
    • 0043242780 scopus 로고
    • Derivation of efficient continuous explicit Runge-Kutta methods
    • B. Owren and M. Zennaro, Derivation of efficient continuous explicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 13 (1992) 1488-1501.
    • (1992) SIAM J. Sci. Statist. Comput. , vol.13 , pp. 1488-1501
    • Owren, B.1    Zennaro, M.2
  • 74
    • 0347742463 scopus 로고
    • ERNY - An explicit Runge-Kutta-Nyström integrator for second order initial value problems
    • Department of Computer Science, University of Toronto
    • P.W. Sharp, ERNY - an explicit Runge-Kutta-Nyström integrator for second order initial value problems, Technical Report No. 199, Department of Computer Science, University of Toronto (1987).
    • (1987) Technical Report No. 199 , vol.199
    • Sharp, P.W.1
  • 75
    • 0347742466 scopus 로고
    • New low order explicit Runge-Kutta pairs
    • Department of Computer Science, University of Toronto
    • P.W. Sharp, New low order explicit Runge-Kutta pairs, Technical Report No. 222, Department of Computer Science, University of Toronto (1989).
    • (1989) Technical Report No. 222 , vol.222
    • Sharp, P.W.1
  • 76
    • 0026216091 scopus 로고
    • Numerical comparison of explicit Runge-Kutta pairs of orders four through eight
    • P.W. Sharp, Numerical comparison of explicit Runge-Kutta pairs of orders four through eight, ACM Trans. Math. Software 17 (1991) 387-409.
    • (1991) ACM Trans. Math. Software , vol.17 , pp. 387-409
    • Sharp, P.W.1
  • 77
    • 0346481891 scopus 로고
    • User guide for ERNY - An explicit Runge-Kutta-Nyström integrator for second order initial value problems
    • Department of Computer Science, University of Toronto
    • P.W. Sharp and J.M. Fine, User guide for ERNY - an explicit Runge-Kutta-Nyström integrator for second order initial value problems, Technical Report No. 198, Department of Computer Science, University of Toronto (1987).
    • (1987) Technical Report No. 198 , vol.198
    • Sharp, P.W.1    Fine, J.M.2
  • 78
    • 38249007701 scopus 로고
    • Some Nyström pairs for the general second order initial value problem
    • P.W. Sharp and J.M. Fine Some Nyström pairs for the general second order initial value problem, J. Comput. Appl. Math. 42 (1992) 279-291.
    • (1992) J. Comput. Appl. Math. , vol.42 , pp. 279-291
    • Sharp, P.W.1    Fine, J.M.2
  • 81
    • 0345850944 scopus 로고
    • Certain equivalent requirements of approximate solutions of x′ = f(t,x)
    • N.F. Stewart, Certain equivalent requirements of approximate solutions of x′ = f(t,x), SIAM J. Numer. Anal. 7 (1970) 256-270.
    • (1970) SIAM J. Numer. Anal. , vol.7 , pp. 256-270
    • Stewart, N.F.1
  • 83
    • 0347111996 scopus 로고
    • Private communication
    • J.H. Verner, Private communication (1976).
    • (1976)
    • Verner, J.H.1
  • 84
    • 0027683528 scopus 로고
    • Differentiable interpolants for higher-order Runge-Kutta methods
    • J.H. Verner, Differentiable interpolants for higher-order Runge-Kutta methods, SIAM J. Numer. Anal. 39 (1993) 1446-1466.
    • (1993) SIAM J. Numer. Anal. , vol.39 , pp. 1446-1466
    • Verner, J.H.1


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