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.

Added October-November 2006

All Pairs have more Accurate Coefficients as of December 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 2. under Journal 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.
Phone:
(778)782-6554

Office:
TASC II, Room 8516, Simon Fraser University

Personal Interests:
Photography, Hiking, Bridge

Teaching

Sept.-Dec., 2001: Math 310: Ordinary Differential Equations
Jan.-April, 2002: Math 154: Calculus I for Biology and Medicine
Jan.-April, 2003: Math 155: Calculus II for Biology and Medicine
Sept.-Dec., 2003: MACM 316: Numerical Analysis I
Jan.-April, 2004: Math 158: Calculus II for the Social Sciences
Sept.-Dec., 2004: Math 152: Calculus II
Jan.-April, 2005: Math 155: Calculus II for Biology and Medicine
Jan.-April, 2006: Math 158: Calculus II for the Social Sciences
Jan.-April, 2007: Math 151: Calculus I for Mathematics and Science
Jan.-April, 2008: Math 157: Calculus I for the Social Sciences

Research interests

Numerical analysis, integration methods for systems of ordinary differential equations

Research groups:


Pacific Institute of Mathematical Sciences
Centre for Scientific Computation

Algorithms


Contemporary Presentations

GLADE: Conference on General Linear Algorithms for Differential Equations, July 2008, Auckland, New Zealand
B-series and TSRK methods based on Gaussian Quadrature
ps format: Abstract
pdf format: Abstract
SciCADE07, July, 2007, St. Malo, France
Numerically Optimal Runge--Kutta Pairs and Interpolants
ps format: Abstract
pdf format: Abstract
SciCADE05, July, 2005, Nagoya, Japan
Order Tests and Derivation of Two-Step Runge--Kutta Pairs of Order 8
ps format: Abstract
pdf format: Abstract
Conference on Numerical Volterra and Delay Equations, May, 2004, Tempe, Arizona
Improved Starting Methods for Two-step Runge--Kutta Methods
ps format: Abstract
pdf format: Abstract
ANODE03, July, 2003, Auckland, New Zealand
Starting Methods for High-order Two-step Runge--Kutta Methods
ps format: Abstract
WODE, December, 2002, Bari, Italy
Why are some Two-step Runge--Kutta Methods Inaccurate?
ps format: Abstract
pdf format: Abstract

Research Publications

Recent Manuscripts

  1. J.H. Verner, Numerically optimal Runge--Kutta pairs with interpolants. Submitted to Numerical Algorithms. November, 2008, 10 pages.

Refereed Journals

(Most recent items first)

  1. 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.
    ps format: Improved Starting Methods for TSRK Methods
    pdf format: Improved Starting Methods for TSRK Methods

  2. 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.
    ps format: Starting Methods for TSRK Methods
    pdf format: Starting Methods for TSRK Methods

  3. T. Macdougall and J.H. Verner, Global error estimators for Order 7,8 Runge--Kutta pairs, Numerical Algorithms 31, (2002) pp. 215--231.
  4. 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
    ps format: Derivation of TSRK Methods
    pdf format: Derivation of TSRK Methods

  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. J.H. Verner, High order explicit Runge--Kutta pairs with low stage order, Applied Numerical Mathematics 22 (1996), pp. 345--357.
  10. J.H. Verner and M. Zennaro, The orders of embedded continuous explicit Runge--Kutta methods, BIT 35 (1995), pp. 406--416.
  11. J.H. Verner and M. Zennaro, Continuous explicit Runge--Kutta methods of order 5, Mathematics of Computation 64 (1995), pp.1123--1146.
  12. J.H. Verner, Strategies for deriving new explicit Runge--Kutta pairs, Annals of Numerical Mathematics 1 (1994), pp. 225--244.
  13. P.W. Sharp and J. H. Verner, Completely imbedded Runge--Kutta pairs. SIAM J. NA 31 (1994), pp. 1169--1190.
  14. J.H. Verner, Differentiable interpolants for high-order Runge--Kutta methods. SIAM J. NA 30 (1993), pp.1446--1466.
  15. J.H. Verner, Some Runge--Kutta formula pairs, SIAM J. NA. 28 (1991), pp. 496--511.
  16. 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.
    pdf format: A comparison of some of RK pairs using DETEST
  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.
    ps format: Classification of RK pairs
    pdf format: Classification of RK pairs

Last modified December 16, 2008.

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