Jim Verner's Refuge for Runge-Kutta Pairs




Interaction in Research and Teaching:


Jim Verner
Ph.D., Edinburgh, October, 1969


My intentions

I have been interested in the derivation of new and better Runge-Kutta algorithms for some time. In particular, I showed that the design of Runge-Kutta pairs by E. Fehlberg could be improved to provide reliable algorithms for treating general initial value problems that might include substantial quadrature components, and constructed a design for generating such algorithms. My intention is to use this site to distribute some of the better algorithms I have derived.

In recent years, I have been developing software for Ordinary Differential Equation Step-by-Step Solution Analysis (ODESSA). Some indication of progress on this project is indicated in items appearing later in this website.

My most recent contribution is a new family of (13,7-8) explicit Runge--Kutta pairs, the derivation of which is based on studying nullspaces of matrices arising from order conditions. While these are not as efficient as pairs previously reported in this website, this family is the first new family that has appeared in many years. A simultaneous study of previously known efficient (13,7-8) pairs has yielded a new pair that is CONSIDERABLY MORE EFFICIENT than any previously published pair, and coefficients of this pair and of corresponding interpolants follow previously reported pairs.

Sets of all coefficients provided in attachments are copyrighted as such by the author. They many not be published for general distribution. They may be used for any research, industrial application or development of software provided that any product arising using any set of coefficients acknowledges this source and includes the URL for this site within the produced item.

Added October-November 2006
Modified November 2008

A "more efficient" Runge--Kutta (6)5 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more efficient RK 6(5) Pair
txt format: Rational coefficients only
txt format: Floating Point coefficients only

A "more robust" Runge--Kutta (6)5 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more robust RK 6(5) Pair
txt format: Rational coefficients only
txt format: Floating Point coefficients only

Added April, 2007 and CORRECTED July, 2024:

A "more efficient" Runge--Kutta (7)6 Pair with corrected Interpolants

Click on ICON at right for stability regions.
txt format: A more efficient RK 7(6) Pair
txt format: Rational and radical coefficients only
txt format: Floating point coefficients only

A "more robust" Runge--Kutta (7)6 Pair with corrected Interpolants

Click on ICON at right for stability regions.
txt format: A more robust RK 7(6) Pair
txt format: Rational and radical coefficients only
txt format: Floating point coefficients only

Added May, 2007:

A "more efficient" Runge--Kutta (8)7 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more efficient RK 8(7) Pair
txt format: Rational coefficients with floating point interpolants
txt format: Floating point coefficients only

A "more robust" Runge--Kutta (8)7 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more robust RK 8(7) Pair
txt format: Rational coefficients with floating point interpolants
txt format: Floating point coefficients only

A "more efficient" Runge--Kutta (9)8 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more efficient RK 9(8) Pair
txt format: Radical coefficients with floating point interpolants
txt format: Floating point coefficients only

A "more robust" Runge--Kutta (9)8 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more robust RK 9(8) Pair
txt format: Radical coefficients with floating point interpolants
txt format: Floating point coefficients only

Three even more efficient procedures than those above. Maximum coefficients are larger.
Added July, 2024:

An even more "efficient" Runge--Kutta (7)6 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A more efficient RK 7(6) Pair
txt format: Rational and radical coefficients only
txt format: Floating point coefficients only

Added May, 2024:

A considerably more "efficient" Runge--Kutta (8)7 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A considerably more efficient RK 8(7) Pair
txt format: Rational coefficients with floating point interpolants
txt format: Floating point coefficients only

Added April, 2024:

A slightly more "efficient" Runge--Kutta (9)8 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A better efficient RK 9(8) Pair
txt format: Radical coefficients with floating point interpolants
txt format: Floating point coefficients only

A Runge--Kutta (8)7 Pair with Minimal Coefficients in the Interpolants

Click on ICON at right for stability regions.
txt format: An RK 8(7) Pair with minimal interpolation coefficients
txt format: Rational coefficients with floating point interpolants
txt format: Floating point coefficients only

Added August, 2023:

Coefficients for a (nearly) optimal nullspace explicit (8)7 Runge--Kutta pair

See recent research paper 1. below.
Click on ICON at right for stability regions.

txt format: A nullspace efficient RK 8(7) Pair

Coefficients for a TSRK6 method with starting methods

See Refereed Journal Article 8. below.
txt format: TSRK6 method with starting methods



Current University Affiliations:

Visitor

Department of Mathematics
Simon Fraser University
8888 University Avenue,
Burnaby, B.C., Canada, V5A 1S6

Professor Emeritus

Department of Mathematics and Statistics
Queen's University at Kingston
Kingston, Ontario, Canada, K7L 3N6

E-mail:
jimverner@shaw.ca


Office:
Science Building, Room K9505.3, Simon Fraser University

Phone:
(778)782-3009

Personal Interests:
Bridge, Photography, Hiking, Canoeing, Skiing

Teaching

Sept. 1963 - April 1964: Royal Military College Kingston, Ontario
Sept. 1964 - July, 1966: Government College, Umuahia, Nigeria
Sept. 1969 - Dec. 2000: Queen's University Department of Mathematics and Statistics
Sept. 2001 - April 2008: Simon Fraser University Department of Mathematics

Research interests

Numerical analysis, integration methods for systems of ordinary differential equations

Research groups:

Pacific Institute of Mathematical Sciences
Centre for Scientific Computation

Algorithms

Trees.TSRK.maple:

A MAPLE code for order conditions and assessment of Two-Step Runge--Kutta (TSRK) methods.
(This code generates the order conditions of arbitrary order for a TSRK method, and provides coefficients of error
espressions which may be used to contrast a TSRK method with any other method for potential efficiency. See below.)

Conference Presentations

  1. Feb. 2023, ANODE2023, University of Auckland, NZ
    Nullspaces yield new explicit Runge--Kutta methods
    pdf format: Abstract

  2. Feb. 2018, ANODE2018, University of Auckland, NZ
    ODESSA - Software for the Analysis of General Linear Methods
    pdf format: Abstract

  3. Jan. 2013, ANODE2013, University of Auckland, NZ
    New Runge--Kutta pairs of lower stage-order
    pdf format: Abstract

  4. July 2011, University of Toronto, Ontario;
    A retrospective on the derivation of Runge--Kutta pairs
    pdf format: Abstract

  5. May 2009, NTNU, Trondheim, Norway;
    June 2009, Department of Mathematics, Laurentian University, Sudbury, Ontario
    B-series and TSRK pairs based on Gaussian quadratures
    pdf format: Abstract

  6. July 2008, GLADE: Conference on General Linear Algorithms for Differential Equations, Auckland, New Zealand
    B-series and TSRK methods based on Gaussian Quadrature
    pdf format: Abstract

  7. July, 2007, SciCADE07, St. Malo, France
    Numerically Optimal Runge--Kutta Pairs and Interpolants
    pdf format: Abstract

  8. July, 2005, SciCADE05, Nagoya, Japan
    Order Tests and Derivation of Two-Step Runge--Kutta Pairs of Order 8
    pdf format: Abstract

  9. May, 2004, Conference on Numerical Volterra and Delay Equations, Tempe, Arizona
    Improved Starting Methods for Two-step Runge--Kutta Methods
    pdf format: Abstract

  10. July,2003, ANODE03, Auckland, New Zealand
    Starting Methods for High-order Two-step Runge--Kutta Methods
    pdf format: Abstract

  11. December, 2002, WODE, Bari, Italy
    Why are some Two-step Runge--Kutta Methods Inaccurate?
    pdf format: Abstract

Research Publications

    Recent Journal Articles

    1. J.H. Verner, Nullspaces yield new explicit Runge--Kutta pairs, Numerical Algorithms, (2024) 28 pages Read only link: https://rdcu.be/dyQ2T
      https://doi.org/10.1007/s11075-023-01731-6
      Springer Preview: Nullspaces yield new explicit Runge--Kutta pairs

    2. A. Cardone, Z. Jackiewicz, J.H. Verner, B. Welfert, Order conditions for general linear methods, J. Applied Mathematics and Computation 290, (2015), pp. 44--64.
      https://doi.org/10.1016/j.cam.2015.04.042
      Abstract: Order conditions for general linear methods

    3. J.H. Verner, Explicit Runge--Kutta pairs with lower stage-order. Numerical Algorithms, 65, (2014) pp. 555--577.
      https://doi.org/10.1007/s11075-013-9783-y
      Abstract: Explicit Runge--Kutta pairs with lower stage-order
      Springer Preview: Explicit Runge--Kutta pairs with lower stage-order

    4. Yiannis Hadjimichael, Colin B. Macdonald, David I. Ketcheson, James H. Verner, Strong stability preserving explicit Runge--Kutta methods of maximal effective order, SIAM J. NA, 51, No. 4 (2013) pp. 2149--2165. https://doi.org/10.1137/120884201
      Abstract: SSP explicit Runge--Kutta methods of maximal effective order
      SIAM NA Preview: SSP explicit Runge--Kutta methods of maximal effective order

    5. Anne Kværnø and Jim Verner, Subquadrature Expansions for TSRK methods. Numerical Algorithms, 59, (2012) pp. 487--504.
      https://doi.org/10.1007/s11075-011-9500-7
      Springer Preview: Subquadrature Expansions for TSRK methods

    6. Anne Kværnø and Jim Verner, Trees.TSRK.maple: a MAPLE code for order conditions and assessment of Two-Step Runge--Kutta (TSRK) methods.
      numeralgo/na32.tgz Netlib Repository (2012).
      Download the zip folder from the library. This folder contains instructions in a README.txt file.
      Numeralgo Library: na32.tgz

    7. J.H. Verner, Numerically optimal Runge--Kutta pairs with interpolants. Numerical Algorithms, 53, (2010) pp. 383--396.
      https://doi.org/10.1007/s11075-009-9290-3
      Abstract: Numerically optimal Runge--Kutta pairs with interpolants
      Springer Preview: Numerically optimal Runge--Kutta pairs with interpolants

    8. J.H. Verner, Improved Starting methods for two-step Runge--Kutta methods of stage-order p-3, Applied Numerical Mathematics, 56, (2006) pp. 388--396.
      https://doi.org/10.1016/j.apnum.2005.04.028
      APNUM Preview: Improved starting methods of stage-order p-3

    9. J.H. Verner, Starting methods for two-step Runge--Kutta methods of stage-order 3 and order 6, J. Computational and Applied Mathematics, 185, (2006)
      pp. 292--307. https://doi.org/10.1016/j.cam.2005.03.012
      Elsevier Preview Starting methods for two-step Runge--Kutta methods of stage-order 3 and order 6

    10. T. Macdougall and J.H. Verner, Global error estimators for order 7,8 Runge--Kutta pairs, Numerical Algorithms 31, (2002) pp. 215--231.
      https://doi.org/10.1023/A:1021190918665
      Num. Alg. Preview Global error estimators for order 7,8 Runge--Kutta pairs.

    11. Z. Jackiewicz and J.H. Verner, Derivation and implementation of two-step Runge--Kutta pairs. Japan Journal of Industrial and Applied Mathematics 19 (2002), pp. 227--248. https://doi.org/10.1007/BF03167454
      JJIAM Preview Derivation and implementation of two-step Runge--Kutta pairs

      A corrected form of this paper is
      pdf: Derivation of TSRK Methods

    12. P.W. Sharp and J.H. Verner, Some extended Bel'tyukov pairs for Volterra integral equations of the the second kind, SIAM Journal on Numerical Analysis 38 (2000), pp. 347--359. https://doi.org/10.1137/S003614299631280X
      SIAM J. NA Preview: Some extended Bel'tyukov pairs for Volterra integral equations of the the second kind

    13. P.W. Sharp and J.H. Verner, Extended explicit Bel'tyukov pairs of orders 4 and 5 for Volterra integral equations of the the second kind, Applied Numerical Mathematics 34 (2000), pp. 261--274. https://doi.org/10.1016/S0168-9274(99)00132-4
      ANM Preview Extended explicit Bel'tyukov pairs of orders 4 and 5 for Volterra integral equations of the the second kind

    14. D.D. Olesky, P. van den Driessche and J.H. Verner, Graphs with the same determinant as a complete graph, Linear Algebra and its Applications 312 (2000),
      pp. 191--195. https://doi.org/10.1016/S0024-3795(00)00114-2
      Elsevier Preview Graphs with the same determinant as a complete graph,

    15. P.W. Sharp and J.H. Verner, Generation of High Order Interpolants for Explicit Runge--Kutta Pairs, AMS Transactions on Mathematical Software 24 (1998), pp.13--29. https://doi.org/10.1145/285861.285863
      AMS ToMS Preview Generation of High Order Interpolants for Explicit Runge--Kutta Pairs

    16. J.H. Verner, High order explicit Runge--Kutta pairs with low stage order, Applied Numerical Mathematics 22 (1996), pp. 345--357.
      https://doi.org/10.1016/S0168-9274(96)00041-4
      ANM Preview High order explicit Runge--Kutta pairs with low stage order

    17. J.H. Verner and M. Zennaro, The orders of embedded continuous explicit Runge--Kutta methods, BIT 35 (1995), pp. 406--416.
      https://doi.org/10.1007/BF01732613
      Springer Preview: The orders of embedded continuous explicit Runge--Kutta methods

    18. J.H. Verner and M. Zennaro, Continuous explicit Runge--Kutta methods of order 5, Mathematics of Computation 64 (1995), pp.1123--1146.
      https://doi.org/10.2307/2153486
      Math. Comp. Preview Continuous explicit Runge--Kutta methods of order 5

    19. J.H. Verner, Strategies for deriving new explicit Runge--Kutta pairs, Annals of Numerical Mathematics 1 (1994), pp. 225--244.
      pdf format: Strategies for deriving new explicit Runge--Kutta pairs

    20. P.W. Sharp and J. H. Verner, Completely imbedded Runge--Kutta pairs. SIAM J. NA 31 (1994), pp. 1169--1190. https://doi.org/10.1137/0731061
      SIAM J. NA Preview Completely imbedded Runge--Kutta pairs

    21. J.H. Verner, Differentiable interpolants for high-order Runge--Kutta methods. SIAM J. NA 30 (1993), pp.1446--1466. https://doi.org/10.1137/0730075
      SIAM J. NA Preview Differentiable interpolants for high-order Runge--Kutta methods

    22. J.H. Verner, Some Runge--Kutta formula pairs, SIAM J. NA. 28 (1991), pp. 496--511. https://doi.org/10.1137/0728027
      SIAM J. NA Preview Some Runge--Kutta formula pairs

    23. J.H. Verner, A contrast of some Runge--Kutta formula pairs, SIAM J. NA. 27 (1990), pp. 1332--1344. https://doi.org/10.1137/0727076
      SIAM J. NA Preview A contrast of some Runge--Kutta formula pairs

    24. J.H. Verner, Families of imbedded Runge--Kutta methods, SIAM J. NA. 16 (1978), pp. 875--885. https://doi.org/10.1137/0716064
      SIAM J. NA. Preview Families of imbedded Runge--Kutta methods

    25. J.H. Verner, Explicit Runge--Kutta methods with estimates of the local truncation error, SIAM J. NA. 15 (1978), pp. 772--790. https://doi.org/10.1137/0715051
      SIAM J. NA. Preview Explicit Runge--Kutta methods with estimates of the local truncation error

    26. A.V. Geramita and J.H. Verner, Orthogonal designs with zero diagonal, Canadian J. Math. 28 (1976), pp. 215--224. https://doi.org/10.4153/CJM-1976-028-3
      Can. J. Math. Preview Orthogonal designs with zero diagonal

    27. G.J. Cooper and J.H. Verner, Some explicit Runge--Kutta methods of high order, SIAM J. NA. 9 (1972), pp. 389--405. https://doi.org/10.1137/0709037
      SIAM J. NA Preview Some explicit Runge--Kutta methods of high order

    28. J.H. Verner, Quadratures for implicit differential equations, SIAM J. NA. 7 (1970), pp. 373--385. https://doi.org/10.1137/0707030
      SIAM J. NA Preview Quadratures for implicit differential equations

    29. J.H. Verner, The order of some implicit Runge--Kutta methods, Numerische Mathematik 13 (1969), pp. 14--23. https://doi.org/10.1007/BF02165270
      Num. Math. Preview The order of some implicit Runge--Kutta methods

    30. J.H. Verner and M.J.M. Bernal, On generalizations of the theory of consistent orderings for successive over--relaxation methods, Numerische Mathematik 12 (1968), pp. 215--222. https://doi.org/10.1007/BF02162914
      Num. Math. Preview On generalizations of the theory of consistent orderings for successive over--relaxation methods

    Conference Proceedings

    1. J.H. Verner, A classification scheme for studying explicit Runge Kutta pairs, Scientific Computing, S. O. Fatunla (editor), Ada and Jane Press, Benin City, Nigeria, 1994, 201-225.
      pdf format: Classification of RK pairs

    2. J.H. Verner, A comparison of some Runge--Kutta formula pairs using DETEST, Computational Ordinary Differential Equations, S.O. Fatunla (editor), Univ. Press PLC, Ibadan, Nigeria, 1991, pp. 271--284.
      pdf format: A comparison of some Runge--Kutta formula pairs using DETEST

    3. J.H. Verner, John Butcher's algebraic theory: Motivation for selecting simplifying conditions, Proceedings of the Ninth Manitoba Conference on Numerical Mathematics, Winnipeg, Manitoba, 1979, pp. 125-155.

    4. J.H. Verner, Selecting a Runge--Kutta method, Proceedings of the Sixth Manitoba Conference on Numerical Mathematics, Winnipeg, Manitoba, 1976, pp. 495--504.

    5. J.H. Verner, On deriving certain hybrid methods, Proc. Manitoba Conf. on Numerical Mathematics, Winnipeg, Manitoba, 1971, pp. 607--626.

    6. J.H. Verner, On deriving explicit Runge--Kutta methods, Proc. on Applications of Numerical Analysis, Dundee, Scotland, 1971, pp. 341--347.

    7. J.H. Verner, Implicit methods for implicit differential equations, Proc. of the Conference on the Numerical Solution of Differential Equations, Dundee, Scotland, 1969, pp. .

    Other Research Reports

    1. P. Chartier, E. Lapôtre, J.H. Verner, Reversible B-series and the derivation of pseudo-symmetric Runge--Kutta methods, INRIA report , Campus de Beaulieu, Rennes, France, 2000, 23 pages.

    2. J.H. Verner, The derivation of high order explicit Runge--Kutta methods, Department of Mathematics Report No. 93, University of Auckland, New Zealand, 1976, 27 pages.

    3. J.H. Verner, Convergence of a finite difference scheme fora second order ordinary differential equation with a singularity, London Institute of Computer Science Report ICS/001/1967 (1967), 19 pages.

    Stories with John

    After I had studied a number of John Butcher's papers during my Ph.D. program, he and I met at my home in Kingston, Canada in December, 1969. At that time he dictated an algorithm in Algol, and I coded it in APL to provide a general procedure for verifying the algebraic order of a Runge--Kutta method. Over many years often at ANODE conferences, we have worked together informally on a variety of Runge--Kutta and other methods. Since the launch of his recent monograph in April, 2021, we have agreed to review some of our interactions, and these are being presented as short stories on his website, jcbutcher.com .

    1. Jim Verner, The first eighth order method, jcbutcher.com/p8

    2. John Butcher and Jim Verner, Our first B-series program, jcbutcher.com/1970

    3. Jim Verner, APL to MAPLE code of RKTest70.APL, jcbutcher.com/node/100

    4. Jim Verner, Deriving new explicit Runge--Kutta pairs using Test21, jcbutcher.com/node/102

      5. On Aug. 29, 2023, the author received notifications of errors in coefficients from two different researchers. The author thanks C.D. for detecting differences in weights for 'effective 6(5)' pairs, and A.J. for detecting nodal errors in 'robust 6(5)' pairs, and reporting errors in 7(6) pairs observed by Peter Stone. Errors in these pairs have been corrected and coefficients satisfy the order conditions stated.

      Please send errors to jimverner@shaw.ca Algorithms and responses may be received on request from this address.

      This website was last modified in July, 2024.