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.

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 They many not be published for general distribution.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 "most efficient" Runge--Kutta (6)5 Pair with Interpolants

Click on ICON at right for stability regions.
txt format: A most efficient RK 56 Pair
txt format: Rational coefficients only
txt format: Floating Point coefficients only

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

Click on ICON at right for stability regions.
txt format: A most robust RK 56 Pair
txt format: Rational coefficients only
txt format: Floating Point coefficients only

Added April, 2007:

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

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

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

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

Added May, 2007:

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

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

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

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

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

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

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

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

Coefficients for a TSRK6 method with starting methods

See Refereed Journal Article 10. below.
txt format: TSRK6 method with starting methods



Current University Affiliations:

Adjunct Professor

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

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

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

  4. 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

  5. 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

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

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

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

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

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


Research Publications

Refereed Journal Articles

  1. 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
  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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.

  10. 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

  11. 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

  12. 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

  13. 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,

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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. of the Conference on Applications of Numerical Analysis, Springer-Verlag Lecture Notes in Mathematics, 228, Dundee, Scotland, 1971, pp. 340--347.

  7. J.H. Verner, Implicit methods for implicit differential equations, Proc. of the Conference on the Numerical Solution of Differential Equations, Springer-Verlag Lecture Notesin Mathematics, 109, Dundee, Scotland, 1969, pp. 261--266.

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.


Last modified Feb. 23, 2022.

Algorithms to and from jimverner@shaw.ca