ODE Tableaus

ODE Tableaus

Explicit Runge-Kutta Methods

Implicit Runge-Kutta Methods

Tableau Methods

stability_region(z,tab::ODERKTableau)

Calculates the stability function from the tableau at z. Stable if <1.

\[r(z) = \frac{\det(I-zA+zeb^T)}{\det(I-zA)}\]

stability_region(tab::ODERKTableau; initial_guess=-3.0)

Calculates the length of the stability region in the real axis.

ODEDEFAULTTABLEAU

Sets the default tableau for the ODE solver. Currently Dormand-Prince 4/5.

Explicit Tableaus

Euler's method.

Ralston's Order 2 method.

Heun's Order 2 method.

Kutta's Order 3 method.

Missing docstring.

Missing docstring for OrdinaryDiffEq.constructBS3. Check Documenter's build log for details.

constructBogakiShampine3()

Constructs the tableau object for the Bogakai-Shampine Order 2/3 method.

Classic RK4 method.

Classic RK4 3/8's rule method.

Runge-Kutta-Fehberg Order 4/3

Runge-Kutta-Fehlberg Order 4/5 method.

constructCashKarp()

Constructs the tableau object for the Cash-Karp Order 4/5 method.

Missing docstring.

Missing docstring for DiffEqDevTools.constructDormandPrince. Check Documenter's build log for details.

Missing docstring.

Missing docstring for OrdinaryDiffEq.constructBS5. Check Documenter's build log for details.

S.N. Papakostas and G. PapaGeorgiou higher error more stable

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.

S.N. Papakostas and G. PapaGeorgiou less stable lower error Strictly better than DP5

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.

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.

Luther and Konen's First Order 5 Some Fifth-Order Classical Runge Kutta Formulas, H.A.Luther and H.P.Konen, Siam Review, Vol. 3, No. 7, (Oct., 1965) pages 551-558.

Luther and Konen's Second Order 5 Some Fifth-Order Classical Runge Kutta Formulas, H.A.Luther and H.P.Konen, Siam Review, Vol. 3, No. 7, (Oct., 1965) pages 551-558.

Luther and Konen's Third Order 5 Some Fifth-Order Classical Runge Kutta Formulas, H.A.Luther and H.P.Konen, Siam Review, Vol. 3, No. 7, (Oct., 1965) pages 551-558.

Runge's First Order 5 method

Lawson's 5th order scheme

An Order Five Runge Kutta Process with Extended Region of Stability, J. Douglas Lawson, Siam Journal on Numerical Analysis, Vol. 3, No. 4, (Dec., 1966) pages 593-597

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.

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

Cassity's Order 5 method

Butcher's First Order 6 method

On Runge-Kutta Processes of High Order, by J. C. Butcher, Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194

Butcher's Second Order 6 method

On Runge-Kutta Processes of High Order, by J. C. Butcher, Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194

Butcher's Third Order 6

On Runge-Kutta Processes of High Order, by J. C. Butcher, Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194

From Verner's Website

TanakaKasugaYamashitaYazaki Order 6 A

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.

constructTanakaKasugaYamashitaYazaki Order 6 B

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.

constructTanakaKasugaYamashitaYazaki Order 6 C

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.

constructTanakaKasugaYamashitaYazaki Order 6 D

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.

Anton Hutas First Order 6 method

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).

Anton Hutas Second Order 6 method

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).

Verner Order 5/6 method

A Contrast of a New RK56 pair with DP56, by Jim Verner, Department of Mathematics. Simon Fraser University, Burnaby, Canada, 2006.

Dormand-Prince Order 5//6 method

P.J. Prince and J. R. Dormand, High order embedded Runge-Kutta formulae, Journal of Computational and Applied Mathematics . 7 (1981), pp. 67-75.

Sharp-Verner Order 5/6 method

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.

Missing docstring.

Missing docstring for DiffEqDevTools.constructVern6. Check Documenter's build log for details.

EXPLICIT RUNGE-KUTFA METHODS WITH ESTIMATES OF THE LOCAL TRUNCATION ERROR

Chummund's First Order 6 method

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 Univeristy)

Chummund's Second Order 6 method

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 Univeristy)

Papakostas's Order 6

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)

Lawson's Order 6

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 620-625.

Tsitouras-Papakostas's Order 6

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.

DormandLockyerMcCorriganPrince Order 6 Global Error Estimation

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.

From Verner's Website

Mikkawy-Eisa Order 6

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.

From Verner's website

EXPLICIT RUNGE-KUTFA METHODS WITH ESTIMATES OF THE LOCAL TRUNCATION ERROR

Completely Imbedded Runge-Kutta Pairs, by P.W.Sharp and J.H.Verner, Siam Journal on Numerical Analysis, Vol.31, No.4. (August 1994) pages 1169-1190.

On the Optimization of Some Nine-Stage Seventh-order Runge-Kutta Method, by M. Tanaka, S. Muramatsu and S. Yamashita, Information Processing Society of Japan, Vol. 33, No. 12 (1992) pages 1512-1526.

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.

On the Optimization of Some Nine-Stage Seventh-order Runge-Kutta Method, by M. Tanaka, S. Muramatsu and S. Yamashita, Information Processing Society of Japan, Vol. 33, No. 12 (1992) pages 1512-1526.

From Verner's website

Missing docstring.

Missing docstring for OrdinaryDiffEq.constructTanYam7. Check Documenter's build log for details.

The Relative Efficiency of Alternative Defect Control Schemes for High-Order Continuous Runge-Kutta Formulas W. H. Enright SIAM Journal on Numerical Analysis, Vol. 30, No. 5. (Oct., 1993), pp. 1419-1445.

constructDormandPrice8()

Constructs the tableau object for the Dormand-Prince Order 6/8 method.

constructRKF8()

Constructs the tableau object for the Runge-Kutta-Fehlberg Order 7/8 method.

Some Explicit Runge-Kutta Methods of High Order, by G. J. Cooper and J. H. Verner, SIAM Journal on Numerical Analysis, Vol. 9, No. 3, (September 1972), pages 389 to 405

Some Explicit Runge-Kutta Methods of High Order, by G. J. Cooper and J. H. Verner, SIAM Journal on Numerical Analysis, Vol. 9, No. 3, (September 1972), pages 389 to 405

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.

The Relative Efficiency of Alternative Defect Control Schemes for High-Order Continuous Runge-Kutta Formulas W. H. Enright SIAM Journal on Numerical Analysis, Vol. 30, No. 5. (Oct., 1993), pp. 1419-1445.

Jim Verner's "Maple" (dverk78)

EXPLICIT RUNGE-KUTFA METHODS WITH ESTIMATES OF THE LOCAL TRUNCATION ERROR

constructDormandPrice8_64bit()

Constructs the tableau object for the Dormand-Prince Order 6/8 method with the approximated coefficients from the paper. This works until below 64-bit precision.

An Eighth Order Runge-Kutta process with Eleven Function Evaluations per Step, by A. R. Curtis, Numerische Mathematik, Vol. 16, No. 3 (1970), pages 268 to 277

Missing docstring.

Missing docstring for OrdinaryDiffEq.constructTsitPap8. Check Documenter's build log for details.

Journal of Applied Mathematics & Decision Sciences, 4(2), 183-192 (2000), "High order explicit Runge-Kutta pairs for ephemerides of the Solar System and the Moon".

Optimized explicit Runge-Kutta pairs of order 9(8), by Ch. Tsitouras, Applied Numerical Mathematics, 38 (2001) 123-134.

Optimized explicit Runge-Kutta pairs of order 9(8), by Ch. Tsitouras, Applied Numerical Mathematics, 38 (2001) 123-134.

From Verner's Webiste

Missing docstring.

Missing docstring for OrdinaryDiffEq.constructVern9. Check Documenter's build log for details.

Verner 1991 First Order 5/6 method

Some Ruge-Kutta Formula Pairs, by J.H.Verner, SIAM Journal on Numerical Analysis, Vol. 28, No. 2 (April 1991), pages 496 to 511.

Verner 1991 Second Order 5/6 method

Some Ruge-Kutta Formula Pairs, by J.H.Verner, SIAM Journal on Numerical Analysis, Vol. 28, No. 2 (April 1991), pages 496 to 511.

From Verner's Webiste

Missing docstring.

Missing docstring for DiffEqDevTools.constructFeagin10. Check Documenter's build log for details.

Feagin10 in Tableau form

Ono10

High-order Explicit Runge-Kutta Formulae, Their uses, and Limitations, A.R.Curtis, J. Inst. Maths Applics (1975) 16, 35-55.

A Runge-Kutta Method of Order 10, E. Hairer, J. Inst. Maths Applics (1978) 21, 47-59.

Tom Baker, University of Teeside. Part of RK-Aid http://www.scm.tees.ac.uk/users/u0000251/research/researcht.htm http://www.scm.tees.ac.uk/users/u0000251/j.r.dormand/t.baker/rk10921m/rk10921m

Missing docstring.

Missing docstring for DiffEqDevTools.constructFeagin12. Check Documenter's build log for details.

On the 25 stage 12th order explicit Runge-Kutta method, by Hiroshi Ono. Transactions of the Japan Society for Industrial and applied Mathematics, Vol. 6, No. 3, (2006) pages 177 to 186

Tableau form of Feagin12

Missing docstring.

Missing docstring for DiffEqDevTools.constructFeagin14. Check Documenter's build log for details.

Tableau form of Feagin14

Implicit Tableaus

Implicit Euler Method

Order 2 Midpoint Method

Order 2 Trapezoidal Rule (LobattoIIIA2)

LobattoIIIA Order 4 method

LobattoIIIB Order 2 method

LobattoIIIB Order 4 method

LobattoIIIC Order 2 method

LobattoIIIC Order 4 method

LobattoIIIC* Order 2 method

LobattoIIIC* Order 4 method

LobattoIIID Order 2 method

LobattoIIID Order 4 method

Gauss-Legendre Order 2.

Gauss-Legendre Order 4.

Gauss-Legendre Order 6.

RadauIA Order 3 method

RadauIA Order 5 method

RadauIIA Order 3 method

RadauIIA Order 5 method