## 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 They many not be published for general distribution.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 1S6Professor 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)

Jan. 2013, ANODE2013, University of Auckland, NZ

New Runge--Kutta pairs of lower stage-order- pdf format: Abstract

July 2011, University of Toronto, Ontario;

A retrospective on the derivation of Runge--Kutta pairs- pdf format: Abstract

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

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

July, 2007, SciCADE07, St. Malo, France

Numerically Optimal Runge--Kutta Pairs and Interpolants- pdf format: Abstract

July, 2005, SciCADE05, Nagoya, Japan

Order Tests and Derivation of Two-Step Runge--Kutta Pairs of Order 8- pdf format: Abstract

May, 2004, Conference on Numerical Volterra and Delay Equations, Tempe, Arizona

Improved Starting Methods for Two-step Runge--Kutta Methods- pdf format: Abstract

July,2003, ANODE03, Auckland, New Zealand

Starting Methods for High-order Two-step Runge--Kutta Methods- pdf format: Abstract

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)

- 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

- 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

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

- 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.tgzNetlib Repository(2012).

- 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

- 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.

- 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.

- T. Macdougall and J.H. Verner, Global error estimators for order 7,8 Runge--Kutta pairs,
Numerical Algorithms31, (2002) pp. 215--231.- Z. Jackiewicz and J.H. Verner, Derivation and implementation of two-step Runge--Kutta pairs.
Japan Journal of Industrial and Applied Mathematics19(2002), pp. 227--248.A corrected form of this paper is

- pdf: Derivation of TSRK Methods

- 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 Analysis38(2000), pp. 347--359.- 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 Mathematics34(2000), pp. 261--274.- D.D. Olesky, P. van den Driessche and J.H. Verner, Graphs with the same determinant as a complete graph,
Linear Algebra and its Applications312(2000), pp. 191--195.- P.W. Sharp and J.H. Verner, Generation of High Order Interpolants for Explicit Runge--Kutta Pairs,
AMS Transactions on Mathematical Software24(1998), pp.13--29.- J.H. Verner, High order explicit Runge--Kutta pairs with low stage order,
Applied Numerical Mathematics22(1996), pp. 345--357.- J.H. Verner and M. Zennaro, The orders of embedded continuous explicit Runge--Kutta methods,
BIT35(1995), pp. 406--416.- J.H. Verner and M. Zennaro, Continuous explicit Runge--Kutta methods of order 5,
Mathematics of Computation64(1995), pp.1123--1146.J.H. Verner, Strategies for deriving new explicit Runge--Kutta pairs,

Annals of Numerical Mathematics1(1994), pp. 225--244.- pdf format: Strategies for deriving new explicit Runge--Kutta pairs
- P.W. Sharp and J. H. Verner, Completely imbedded Runge--Kutta pairs.
SIAM J. NA31(1994), pp. 1169--1190.- J.H. Verner, Differentiable interpolants for high-order Runge--Kutta methods.
SIAM J. NA30(1993), pp.1446--1466.- J.H. Verner, Some Runge--Kutta formula pairs,
SIAM J. NA.28(1991), pp. 496--511.- J.H. Verner, A contrast of some Runge--Kutta formula pairs,
SIAM J. NA.27(1990), pp. 1332--1344.

## Conference Proceedings

- 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.- 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)