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numeric::butcher -- Butcher parameters of Runge-Kutta schemes

Introduction

numeric::butcher(method) returns the Butcher parameters of the Runge-Kutta scheme named method.

Call(s)

numeric::butcher(method)

Parameters

method - name of the Runge-Kutta scheme, one of EULER1, RKF43, RK4, RKF34, RKF54a, RKF54b, DOPRI54, CK54, RKF45a, RKF45b, DOPRI45, CK45, BUTCHER6, RKF87, RKF78.

Returns

A list [s,c,a,b1,b2,order1,order2] is returned.

Related Functions

numeric::odesolve

Details

Example 1

The Butcher data of the classical 4 stage, 4th order Runge-Kutta scheme are:

>> numeric::butcher(RK4) 
      --                        +-                -+
      |                         |   0,   0,  0, 0  |
      |                         |                  |
      |     +-              -+  |  1/2,  0,  0, 0  |
      |  4, | 0, 1/2, 1/2, 1 |, |                  |,
      |     +-              -+  |   0,  1/2, 0, 0  |
      |                         |                  |
      |                         |   0,   0,  1, 0  |
      --                        +-                -+
      
                                                              --
                                                               |
                                                               |
         +-                  -+  +-                  -+        |
         | 1/6, 1/3, 1/3, 1/6 |, | 1/6, 1/3, 1/3, 1/6 |, 4, 4  |
         +-                  -+  +-                  -+        |
                                                               |
                                                               |
                                                              --

Note that the weights b1 and b2 coincide: this classical method does not provide an embedded pair.

Example 2

The Butcher data of the embedded Runge-Kutta-Fehlberg pair RKF34 of orders 3 and 4 are:

>> [s, c, a, b1, b2, order1, order2] := numeric::butcher(RKF34): 

The number of stages s of the 4th order subprocess is 5, the abscissae c and the matrix a are given by:

>> s, c, a
         +-                   -+
      5, | 0, 1/4, 4/9, 6/7, 1 |,
         +-                   -+
      
         +-                                      -+
         |    0,       0,        0,       0,   0  |
         |                                        |
         |   1/4,      0,        0,       0,   0  |
         |                                        |
         |   4/81,   32/81,      0,       0,   0  |
         |                                        |
         |  57/98, -432/343, 1053/686,    0,   0  |
         |                                        |
         |   1/6,      0,      27/52,  49/156, 0  |
         +-                                      -+

Using these parameters with the weights

>> b1, b2
      +-                        -+
      | 1/6, 0, 27/52, 49/156, 0 |,
      +-                        -+
      
         +-                                  -+
         | 43/288, 0, 243/416, 343/1872, 1/12 |
         +-                                  -+

yields a numerical scheme of order 3 or 4, respectively:

>> order1, order2
                                   3, 4
>> delete s, c, a, b1, b2, order1, order2:

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