• Main page for Runge-Kutta schemes

Order conditions for an order 6 Runge-Kutta scheme - RKcoeff6a.mws

• Order conditions for an order 6 Runge-Kutta scheme and their associated rooted trees.

Derivation of 7 stage, order 6 Runge-Kutta schemes - RKcoeff6b.mws

• Two schemes of John Butcher, Chammud's scheme etc.
  See: (1)  "On Runge-Kutta Processes of High Order", by J. C. Butcher,
            Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194.
          (2)  "A three-dimensional family of seven-step Runge-Kutta methods of order 6", by G. M. Chammud (Hammud),
           Numerical Methods and programming, 2001, Vol.2, 2001, pages 159-166
            (Advanced Computing Scientific journal published by the Research Computing Center of the Lomonosov Moscow State University)
          (3)  "Optimal first- to sixth-order accurate Runge-Kutta schemes", by E. A. Alshina, E. M. Zaks and N. N. Kalitkin.
             Computational Mathematics and Mathematical Physics, Vol. 48, No. 3, pages 395-405.
   
• Coefficients for the schemes and some properties
  RKcoeff6b_1.pdf     RKcoeff6b_2.pdf     RKcoeff6b_3.pdf     RKcoeff6b_4.pdf     RKcoeff6b_5.pdf     RKcoeff6b_6.pdf     RKcoeff6b_7.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes - RKcoeff6c.mws

• Two schemes of Anton Huta and related schemes.
  See: (1)  "Une amélioration de la méthode de Runge-Kutta-Nyström pour la résolution numérique des équations différentielles du premièr ordre", by Anton Huta,
             Acta Fac. Nat. Univ. Comenian Math., Vol. 1, pages 201-224 (1956).
          (2)  "Contribution à la formule de sixième ordre dans la méthode de Runge-Kutta-Nyström", by Anton Huta,
             Acta Fac.Nat. Univ. Comenian Math., Vol. 2, pages 21-24 (1957).
   
• Coefficients for the schemes and some properties
  RKcoeff6c_1.pdf     RKcoeff6c_2.pdf     RKcoeff6c_3.pdf     RKcoeff6c_4.pdf     RKcoeff6c_5.pdf

Derivation of 7 stage, order 6 Runge-Kutta schemes - RKcoeff6d.mws

• A general stage-order 2 scheme with examples by John Butcher etc.
  See: "On Runge-Kutta Processes of High Order", by J. C. Butcher,
            Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194.
   
• Coefficients for the schemes and some properties
  RKcoeff6d_1.pdf     RKcoeff6d_2.pdf     RKcoeff6d_3.pdf     RKcoeff6d_4.pdf     RKcoeff6d_5.pdf

Derivation of 8 stage, combined order 5 and 6 Runge-Kutta schemes - RKcoeff6e.mws

• Schemes of Prince-Dormand and Jim Verner etc.
  See: (1)  "High order embedded Runge-Kutta formulae", by P.J. Prince and J. R. Dormand,
            Journal of Computational and Applied  Mathematics, 7 (1981), pp. 67-75.
          (2)  A Contrast of a New RK56 pair with DP56, by Jim Verner,
             Department of Mathematics. Simon Fraser University, Burnaby, Canada,  2006.
   
• Coefficients for the schemes and some properties
  RKcoeff6e_1.pdf     RKcoeff6e_2.pdf     RKcoeff6e_3.pdf     RKcoeff6e_4.pdf     RKcoeff6e_5.pdf     RKcoeff6e_6.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9 stage order 5 FSAL embedded scheme - RKcoeff6f.mws

• A scheme due to Philip Sharp and Jim Verner together with similar schemes.
  See: "Completely Imbedded Runge-Kutta Pairs", by P. W. Sharp and J. H. Verner,
           Siam Journal on Numerical Analysis, Vol. 31, No. 4. (Aug., 1994), pages 1169 to 1190.
   
• Coefficients for the schemes and some properties
  RKcoeff6f_1.pdf     RKcoeff6f_2.pdf     RKcoeff6f_3.pdf     RKcoeff6f_4.pdf     RKcoeff6f_5.pdf     RKcoeff6f_6.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9 stage order 5 FSAL embedded scheme - RKcoeff6g.mws

• A scheme due to S.N. Papakostas together with similar schemes.
  See: "On Phase-Fitted modified Runge-Kutta Pairs of order 6(5)", by Ch. Tsitouras and I. Th. Famelis,
          International Conference of Numerical Analysis and Applied Mathematics, Crete, (2006) Extended Abstract.
  Note: The scheme described in this paper was originally constructed by S.N. Papakostas.
   
• Coefficients for the schemes and some properties
  RKcoeff6g_1.pdf     RKcoeff6g_2.pdf     RKcoeff6g_3.pdf     RKcoeff6g_4.pdf     RKcoeff6g_5.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9 stage order 5 FSAL embedded scheme - RKcoeff6h.mws

• A scheme due to Ch. Tsitouras together with similar schemes.
  See: A parameter study of explicit Runge-Kutta pairs of orders 6(5), by Ch. Tsitouras,
         Applied Mathematics Letters, Vol. 11, pages 65 to 69, (1998).
   
• Coefficients for the schemes and some properties
  RKcoeff6h_1.pdf     RKcoeff6h_2.pdf     RKcoeff6h_3.pdf     RKcoeff6h_4.pdf     RKcoeff6h_5.pdf     RKcoeff6h_6.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9 stage order 5 FSAL embedded scheme - RKcoeff6k.mws

• Schemes with specific stability properties.
   
• Coefficients for the schemes and some properties
  RKcoeff6k_1.pdf     RKcoeff6k_2.pdf     RKcoeff6k_3.pdf     RKcoeff6k_4.pdf

Derivation of 7 stage, order 6 Runge-Kutta schemes, each with an 8th stage to give an embedded order 5 scheme - RKcoeff6m.mws

• Schemes similar to those in the family described by Luther.
  See: "An Explicit Sixth Order Runge-Kutta Formula", by H.A.Luther,
          Mathematics of Computation, Vol 22, No. 102, April 1968, pages 434-436.
   
• Coefficients for the schemes and some properties
  RKcoeff6m_1.pdf     RKcoeff6m_2.pdf     RKcoeff6m_3.pdf     RKcoeff6m_4.pdf     RKcoeff6m_5.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9 stage order 5 FSAL embedded scheme - RKcoeff6n.mws

• Stage-order 2 schemes of Papakostas, Tsitouras and Papageorgiou together with similar schemes.
  See: "A General Family of Explicit Runge-Kutta Pairs of Order 6(5)", by S. N. Papakostas, Ch. Tsitouras and G. Papageorgiou,
          Siam Journal of Numerical Analysis, Volume 33, Number 3, June 1996, pages 917 to 936.
   
• Coefficients for the schemes and some properties
  RKcoeff6n_1.pdf     RKcoeff6n_2.pdf     RKcoeff6n_3.pdf     RKcoeff6n_4.pdf     RKcoeff6n_5.pdf     RKcoeff6n_6.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9 stage order 5 FSAL embedded scheme - RKcoeff6p.mws

• A stage-order 3 scheme of Lockyer, McCorrigan and Prince together with similar schemes.
  See: "Global Error estimation with Runge-Kutta triples", by J.R.Dormand, M.A.Lockyer, N.E.McCorrigan and P.J.Prince,
          Computers and Mathematics with Applications, 18 (1989) pages 835-846.
   
• Coefficients for the schemes and some properties
  RKcoeff6p_1.pdf     RKcoeff6p_2.pdf     RKcoeff6p_3.pdf     RKcoeff6p_4.pdf     RKcoeff6p_5.pdf

Derivation of 7 stage, order 6 Runge-Kutta schemes - RKcoeff6q.mws

• Stage-order 3 schemes of Lawson and Tsitouras-Papakostas etc.
  See: (1)  "An Order 6 Runge-Kutta Process with an Extended Region of Stability, by J. D. Lawson,
            Siam Journal on Numerical Analysis, Vol. 4, No. 4 (Dec. 1967), pages 620 to 625.
          (2)  "Cheap Error Estimation for Runge-Kutta methods", by Ch. Tsitouras and S.N. Papakostas,
            Siam Journal on Scientific Computing, Vol. 20, Issue 6, Nov 1999.
   
• Coefficients for the schemes and some properties
  RKcoeff6q_1.pdf     RKcoeff6q_2.pdf     RKcoeff6q_3.pdf     RKcoeff6q_4.pdf     RKcoeff6q_5.pdf     RKcoeff6q_6.pdf

Derivation of 7 stage, order 6 Runge-Kutta schemes, each with an 8th stage to give an embedded order 5 scheme - RKcoeff6r.mws

• Two schemes of Tanaka, Kasuga, Yamashita and Yazaki etc.
  See: "On the Optimization of Some Seven-stage Sixth-order Runge-Kutta Methods", by M. Tanaka, K. Kasuga, S. Yamashita and H. Yazaki,
          Journal of the Information Processing Society of Japan, Vol. 33, No. 8 (1992), pages 993 to 1005.
   
• Coefficients for the schemes and some properties
  RKcoeff6r_1.pdf     RKcoeff6r_2.pdf     RKcoeff6r_3.pdf     RKcoeff6r_4.pdf     RKcoeff6r_5.pdf     RKcoeff6r_6.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9th stage to give an embedded order 5 scheme - RKcoeff6s.mws

• Six schemes of Tanaka, Kasuga, Yamashita and Yazaki etc.
  See: "On the Optimization of Some Eight-stage Sixth-order Explicit Runge-Kutta Method", by M. Tanaka, K. Kasuga, S. Yamashita and H. Yazaki,
          Journal of the Information Processing Society of Japan, Vol. 34, No. 1 (1993), pages 62 to 74.
   
• Coefficients for the schemes and some properties
  RKcoeff6s_1.pdf     RKcoeff6s_2.pdf     RKcoeff6s_3.pdf     RKcoeff6s_4.pdf     RKcoeff6s_5.pdf     RKcoeff6s_6.pdf

Derivation of 8 stage, order 6 Runge-Kutta schemes, each with a 9th stage to give an embedded order 5 scheme - RKcoeff6t.mws

• A minor modification of a scheme of Tanaka, Kasuga, Yamashita and Yazaki etc.
   
• Coefficients for the schemes and some properties
  RKcoeff6t_1.pdf     RKcoeff6t_2.pdf     RKcoeff6t_3.pdf

Derivation of 7 stage, combined order 6 and 4 Runge-Kutta schemes - RKcoeff6u.mws

• A scheme of El-Mikkawy and Eisa together with a scheme of Tsitouras and S.N. Papakostas etc.
  See: (1) "A general four-parameter non-FSAL embedded Runge–Kutta algorithm of orders 6 and 4 in seven stages,"
            by M.E.A. El-Mikkawy and M.M.M. Eisa,
            Applied Mathematics and Computation, Vol. 143, No. 2,  (2003) pages 259 to 267.
          (2)  Cheap Error Estimation for Runge-Kutta methods, by Ch. Tsitouras and S.N. Papakostas,
            Siam Journal on Scientific Computing, Vol. 20, Issue 6, Nov 1999.
   
• Coefficients for the schemes and some properties
  RKcoeff6u_1.pdf     RKcoeff6u_2.pdf     RKcoeff6u_3.pdf     RKcoeff6u_4.pdf     RKcoeff6u_5.pdf     RKcoeff6u_6.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

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