Jim Verner's Refuge for Runge-Kutta Pairs





Interaction in Research and Teaching:


Jim Verner
Ph.D., Edinburgh


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 realiable 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.

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 They many not be published for general distribution.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 item 6. under Research Publications 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:
jverner@pims.math.ca
(More accurate coefficients for Runge--Kutta Pairs are available on request.)

Office:
TASC 2, Room 8516, Simon Fraser University

Phone:
(778)782-6554

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.)

Contemporary Presentations

(Most recent items first)
  1. Jan. 2013, ANODE2013, University of Auckland, NZ
    New Runge--Kutta pairs of lower stage-order
    pdf format: Abstract

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

  3. 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

  4. 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

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

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

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

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

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


Research Publications

Recent Manuscripts

  • A. Cardone, Z. Jackiewicz, J.H. Verner, B. Welfert, Order conditions for general linear methods, Submitted to Applied Mathematics and Computation , July, 2013, 39 pages. Developed while the first and third authors were visitors at the University of Arizona, Tempe, AZ during the spring of 2013.
    Abstract: Order conditions for general linear methods

    Available by email as a .pdf file on request from the first author.

    Refereed Journal Articles

    (Most recent items first)
    1. J.H. Verner, Explicit Runge--Kutta pairs with lower stage-order. Numerical Algorithms, (2013) to appear.
      Abstract: Explicit Runge--Kutta pairs with lower stage-order
      The final publication is available at link.springer.com under URL: DOI: 10.1007/s11075-013-9783-y
      Springer Link: Explicit Runge--Kutta pairs with lower stage-order
      pdf: Explicit Runge--Kutta pairs with lower stage-order

    2. 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.
      Abstract: SSP explicit Runge--Kutta methods of maximal effective order
      The final publication is available at www.siam.org/journals/sinum/51-4/88420.html under URL: DOI: 10.1137/120884201

      SIAM NA Link: SSP explicit Runge--Kutta methods of maximal effective order

    3. Anne Kværnø and Jim Verner, Subquadrature Expansions for TSRK methods. Numerical Algorithms,59, (2012) pp. 487--504.
      pdf: Subquadrature Expansions for TSRK methods

    4. 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).

    5. J.H. Verner, Numerically optimal Runge--Kutta pairs with interpolants. Numerical Algorithms, 53, (2010) pp. 383--396. 10.1007/s11075-009-9290-3
      Abstract: Numerically optimal Runge--Kutta pairs with interpolants
      The final publication is available at link.springer.com under URL: DOI: 10.1007/s11075-009-9290-3
      Springer Link: Numerically optimal Runge--Kutta pairs with interpolants
      pdf: Numerically optimal Runge--Kutta pairs with interpolants

    6. J.H. Verner, Improved Starting methods for two-step Runge--Kutta methods of stage-order p-3, Applied Numerical Mathematics, 10, (2006) pp. 388--396.
      pdf: Improved starting methods for TSRK methods

    7. 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.
      pdf: Starting methods for TSRK methods

    8. T. Macdougall and J.H. Verner, Global error estimators for order 7,8 Runge--Kutta pairs, Numerical Algorithms 31, (2002) pp. 215--231.
    9. 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.
      A corrected form of this paper is
      pdf: Derivation of TSRK Methods

    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. J.H. Verner, High order explicit Runge--Kutta pairs with low stage order, Applied Numerical Mathematics 22 (1996), pp. 345--357.
    15. J.H. Verner and M. Zennaro, The orders of embedded continuous explicit Runge--Kutta methods, BIT 35 (1995), pp. 406--416.
    16. J.H. Verner and M. Zennaro, Continuous explicit Runge--Kutta methods of order 5, Mathematics of Computation 64 (1995), pp.1123--1146.
    17. 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

    18. P.W. Sharp and J. H. Verner, Completely imbedded Runge--Kutta pairs. SIAM J. NA 31 (1994), pp. 1169--1190.
    19. J.H. Verner, Differentiable interpolants for high-order Runge--Kutta methods. SIAM J. NA 30 (1993), pp.1446--1466.
    20. J.H. Verner, Some Runge--Kutta formula pairs, SIAM J. NA. 28 (1991), pp. 496--511.
    21. J.H. Verner, A contrast of some Runge--Kutta formula pairs, SIAM J. NA. 27 (1990), pp. 1332--1344.

    Conference Proceedings

    1. 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, 1992, pp. 271-284.
    2. 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

    Earlier articles by the author extending back to 1967 will be listed here in a subsequent update.


    Last modified November 21, 2013.

    Algorithms to and from jverner@pims.math.ca (Jim Verner)