Maple worksheets on the derivation of Runge-Kutta schemes |

Numerical methods topics:

- Introduction - errors
- Root-finding
- Interpolation
- Numerical integration
- 1st order differential equations
- 2nd order differential equations
- Linear systems
- Finite difference methods
- The Duffing equation
- Approximation of functions
- The numerical evaluation of mathematical functions
- Special inverse functions
- The derivation of Runge-Kutta schemes
- Interpolation for Runge-Kutta schemes

The following Maple worksheets can be downloaded.

They are all compatible with

Classic Worksheet Maple 10.

Derivation of order 3 and 4 Runge-Kutta schemes using Taylor series- RKcoeff1.mws

- Derivation of a system of equations for calculating order 3 Runge-Kutta coefficients.
- The classical order 3 Runge-Kutta method.
- Derivation of a system of equations for calculating order 4 Runge-Kutta coefficients.
- The classical order 4 Runge-Kutta method.

Procedures related to the construction of Runge-Kutta schemes- RKcoeff_draw_tree.mws

- Procedures related to the construction and drawing of rooted trees along with their associated order conditions:

OrderConditionTrees,AllTrees,SimpleTrees,AllSimpleTrees,TreeOrderCondition,,

TreeOrderCondition0TreeErrorTerm,TreeErrorCondition0,DrawTree.- Procedures related to the construction of Runge-Kutta schemes via rooted trees:

OrderConditions,RowSumConditions,StageOrderConditions,PrincipalErrorConditions,,

PrincipalErrorTermsQuadratureConditions,SubquadratureConditions,SimpleOrderConditions,SimpleSubquadratureConditions.- These procedures are based on Maple code from "Runge-Kutta Methods, Trees, and Maple",

by Folkmar Bornemann, Selcuk Journal of Applied Mathematics, Vol. 2, No. 1, pp. 3-15, 2001.

Derivation of order 3 Runge-Kutta schemes- RKcoeff3.mws

- The classical order 3 scheme.
- Other examples.

Derivation of 4 stage, order 4 Runge-Kutta schemes- RKcoeff4a.mws

- The classical order 4 Runge-Kutta method.
- The 3/8 order 4 Runge-Kutta method.
- Gill's order 4 Runge-Kutta method.
- Dormand's order 4 Runge-Kutta method.
- Other examples.

Derivation of 4 stage, order 4 Runge-Kutta schemes- RKcoeff4b.mws

- Ralston's method and two methods by King.

See: (1) "Runge-Kutta Methods with Minimum Error Bounds", by Anthony Ralston,

Mathematics of Computation, Vol. 16, 1962, pages 431 to 437.

(2) "Runge-Kutta Methods with Constrained Minimum Error Bounds", by Richard King,

Mathematics of Computation, Vol. 20, 1966, pages 386 to 391.- Prince's 5 stage, order 4 Runge-Kutta method.
- Other examples.

Higher order Runge-Kutta schemes

The absolute stability region of an explicit Runge-Kutta method- RKcoeff_stability.mws

- Details concerning the stability region of an explicit Rune-Kutta method.
- The stability radius and the imaginary axis inclusion of a Runge-Kutta method.
- The procedure
StabilityFunctioncan be used to construct the stability function of a an explicit Runge-Kutta method.

Evaluating order conditions of an explicit Runge-Kutta method using matrices- RKcoeff_matrix.mws

- The conversion procedure
convert/Matrix_formenables an order condition to be evaluated by matrix multiplication.

This may be useful for some of the high order schemes.

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