Maple worksheets on the derivation of order 5 Runge-Kutta schemes |

Derivation of 6 stage, order 5 Runge-Kutta schemes- RKcoeff5a.mws

- Order conditions for an order 5 Runge-Kutta scheme and their associated rooted trees.

- Schemes of Runge, Cassity, Butcher and Fehlberg

- Coefficients for the schemes and some properties

RKcoeff5a_1.pdf RKcoeff5a_2.pdf RKcoeff5a_3.pdf RKcoeff5a_4.pdf RKcoeff5a_5.pdf RKcoeff5a_6.pdf

Derivation of 6 stage, order 5 Runge-Kutta schemes- RKcoeff5b.mws

- A scheme of Lawson, two schemes of Luther and Konen, a scheme of Kutta etc.

- Coefficients for the schemes and some properties

RKcoeff5b_1.pdf RKcoeff5b_2.pdf RKcoeff5b_3.pdf RKcoeff5b_4.pdf RKcoeff5b_5.pdf RKcoeff5b_6.pdf

Derivation of 6 stage, combined order 4 and 5 Runge-Kutta schemes- RKcoeff5c.mws

- Stage-order 3 schemes of Prince-Dormand, Fehlberg and Cash-Karp

- Coefficients for the schemes and some properties

RKcoeff5c_1.pdf RKcoeff5c_2.pdf RKcoeff5c_3.pdf RKcoeff5c_4.pdf RKcoeff5c_5.pdf RKcoeff5c_6.pdf

Derivation of 6 stage, order 5 Runge-Kutta schemes, each with a 7 stage, order 4 FSAL embedded scheme- RKcoeff5d.mws

- A stage-order 3 scheme of Prince and Dormand together with schemes that satisfy the same principal error conditions.

See: "A family of embedded Runge-Kutta formulae", by J. R. Dormand and P. J. Prince,

Journal of Computational and Applied Mathematics, Vol 6, No. 1, 1980, pages 19 to 26.

There are also two stage-order 3 schemes that do not satisfy any principal error conditions.

- Coefficients for the schemes and some properties

RKcoeff5d_1.pdf RKcoeff5d_2.pdf RKcoeff5d_3.pdf RKcoeff5d_4.pdf

RKcoeff5d_5.pdf RKcoeff5d_6.pdf RKcoeff5d_7.pdf RKcoeff5d_8.pdf

Derivation of 6 stage, combined order 4 and 5 Runge-Kutta schemes- RKcoeff5e.mws

- A stage-order 2 scheme of Papakostas and Papageorgiou together with other stage-order 2 schemes.

See: "A Family of Fifth-order Runge-Kutta Pairs", by S.N. Papakostas and G. PapaGeorgiou,

Mathematics of Computation,Volume 65, Number 215, July 1996, Pages 1165-1181.

- Coefficients for the schemes and some properties

RKcoeff5e_1.pdf RKcoeff5e_2.pdf RKcoeff5e_3.pdf RKcoeff5e_4.pdf RKcoeff5e_5.pdf

Derivation of 6 stage, order 5 Runge-Kutta schemes, each with a 7 stage, order 4 FSAL embedded scheme- RKcoeff5f.mws

- A stage-order 2 scheme of Papakostas and Papageorgiou together with other stage-order 2 schemes.

See: "A Family of Fifth-order Runge-Kutta Pairs", by S.N. Papakostas and G. PapaGeorgiou,

Mathematics of Computation,Volume 65, Number 215, July 1996, Pages 1165-1181.

- Coefficients for the schemes and some properties

RKcoeff5f_1.pdf RKcoeff5f_2.pdf RKcoeff5f_3.pdf RKcoeff5f_4.pdf RKcoeff5f_5.pdf

Derivation of 6 stage, order 5 Runge-Kutta schemes, each with a 7 stage, order 4 FSAL embedded scheme- RKcoeff5g.mws RKcoeff5h.mws

- Various stage-order 2 schemes constructed using the algorithm of Papakostas and Papageorgiou.

See: "A Family of Fifth-order Runge-Kutta Pairs", by S.N. Papakostas and G. PapaGeorgiou,

Mathematics of Computation,Volume 65, Number 215, July 1996, Pages 1165-1181.

- Coefficients for the schemes and some properties

RKcoeff5g_1.pdf RKcoeff5g_2.pdf RKcoeff5g_3.pdf RKcoeff5g_4.pdf

RKcoeff5h_1.pdf RKcoeff5h_2.pdf RKcoeff5h_3.pdf RKcoeff5h_4.pdf

- Stage-order 2 schemes that satisfy various groups of principal error conditions.

See: "REU Numerical Analysis Project on the Design and Optimization of Explicit Runge-Kutta Formulas",

by Stephen Dupal and Michael Yoshizawa, 2007.

- Coefficients for the schemes and some properties

RKcoeff5k_1.pdf RKcoeff5k_2.pdf RKcoeff5k_3.pdf RKcoeff5k_4.pdf RKcoeff5k_5.pdf RKcoeff5k_6.pdf

- A stage-order 1 scheme of Tsitouras together with other stage-order 1 schemes.

See: "Runge–Kutta pairs of orders 5(4) using the minimal set of simplifying assumptions,"

by Ch. Tsitouras, TEI of Chalkis, Dept. of Applied Sciences, GR34400, Psahna, Greece.

- Coefficients for the schemes and some properties

RKcoeff5m_1.pdf RKcoeff5m_2.pdf RKcoeff5m_3.pdf RKcoeff5m_4.pdf RKcoeff5m_5.pdf RKcoeff5m_6.pdf

RKcoeff5n_1.pdf RKcoeff5n_2.pdf RKcoeff5n_3.pdf RKcoeff5n_4.pdf RKcoeff5n_5.pdf RKcoeff5n_6.pdf

Derivation of 7 stage, combined order 4 and 5 Runge-Kutta schemes- RKcoeff5p.mws

- A scheme of Bogacki and Shampine together with other schemes.

See: An Efficient Runge-Kutta (4,5) Pair by P.Bogacki and L.F.Shampine

Computers and Mathematics with Applications, Vol. 32, No. 6, 1996, pages 15 to 28

- Coefficients for the schemes and some properties

RKcoeff5p_1.pdf RKcoeff5p_2.pdf RKcoeff5p_3.pdf RKcoeff5p_4.pdf RKcoeff5p_6.pdf RKcoeff5p_7.pdf

Derivation of 7 stage, combined order 4 and 5 Runge-Kutta schemes- RKcoeff5q.mws

- A scheme due to P. Sharp and E. Smart together with similar schemes.

See: Explicit Runge-Kutta Pairs with One More Derivative Evaluation than the Minimum, by P.W.Sharp and E.Smart,

Siam Journal of Scientific Computing, Vol. 14, No. 2, pages. 338-348, March 1993.

- Coefficients for the schemes and some properties

RKcoeff5q_1.pdf RKcoeff5q_2.pdf RKcoeff5q_3.pdf RKcoeff5q_4.pdf RKcoeff5q_5.pdf RKcoeff5q_6.pdf RKcoeff5q_7.pdf

Proceduresrelated to the construction of Runge-Kutta schemes- butcher.zip

Procedures for root-finding and determining minimum points- roots.zip

Procedures for numerical integration- intg.zip