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Maple worksheets on the derivation of order 5 Runge-Kutta schemes |
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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.pdfDerivation 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.pdfDerivation 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.pdfDerivation 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.pdfDerivation 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.pdfDerivation 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
Derivation of 6 stage, order 5 Runge-Kutta schemes, each with a 7 stage, order 4 FSAL embedded scheme - RKcoeff5k.mws
- 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
Derivation of 6 stage, order 5 Runge-Kutta schemes, each with a 7 stage, order 4 FSAL embedded scheme - RKcoeff5m.mws RKcoeff5n.mws
- 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.pdfDerivation 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.pdfDerivation 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
Procedures related 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
