1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
|
########################################################################
##
## Copyright (C) 2013-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{fcn}, @var{t}, @var{x}, @var{dt})
## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{fcn}, @var{t}, @var{x}, @var{dt}, @var{options})
## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{fcn}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals})
## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{fcn}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals}, @var{t_next})
## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}] =} runge_kutta_23 (@dots{})
## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}, @var{k_vals_out}] =} runge_kutta_23 (@dots{})
##
## This function can be used to integrate a system of ODEs with a given initial
## condition @var{x} from @var{t} to @var{t+dt}, with the Bogacki-Shampine
## method of third order. For the definition of this method see
## @url{http://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods}.
##
## @var{fcn} is a function handle that defines the ODE: @code{y' = f(tau,y)}.
## The function must accept two inputs where the first is time @var{tau} and
## the second is a column vector of unknowns @var{y}.
##
## @var{t} is the first extreme of integration interval.
##
## @var{x} is the initial condition of the system..
##
## @var{dt} is the timestep, that is the length of the integration interval.
##
## The optional fourth argument @var{options} specifies options for the ODE
## solver. It is a structure generated by @code{odeset}. In particular it
## contains the field @var{funarguments} with the optional arguments to be used
## in the evaluation of @var{fcn}.
##
## The optional fifth argument @var{k_vals_in} contains the Runge-Kutta
## evaluations of the previous step to use in a FSAL scheme.
##
## The optional sixth argument @var{t_next} (@code{t_next = t + dt}) specifies
## the end of the integration interval. The output @var{x_next} s the higher
## order computed solution at time @var{t_next} (local extrapolation is
## performed).
##
## Optionally the functions can also return @var{x_est}, a lower order solution
## for the estimation of the error, and @var{k_vals_out}, a matrix containing
## the Runge-Kutta evaluations to use in a FSAL scheme or for dense output.
##
## @seealso{runge_kutta_45_dorpri}
## @end deftypefn
function [t_next, x_next, x_est, k] = runge_kutta_23 (fcn, t, x, dt,
options = [],
k_vals = [],
t_next = t + dt)
persistent a = [0 0 0;
1/2 0 0;
0 3/4 0];
persistent b = [0, 1/2, 3/4, 1];
persistent c = [2/9, 1/3, 4/9];
persistent c_prime = [7/24, 1/4, 1/3, 1/8];
s = t + dt * b;
cc = dt * c;
aa = dt * a;
k = zeros (rows (x), 4);
if (! isempty (options)) # extra arguments for function evaluator
args = options.funarguments;
else
args = {};
endif
if (! isempty (k_vals)) # k values from previous step are passed
k(:,1) = k_vals(:,end); # FSAL property
else
k(:,1) = feval (fcn, t, x, args{:});
endif
k(:,2) = feval (fcn, s(2), x + k(:,1) * aa(2, 1).', args{:});
k(:,3) = feval (fcn, s(3), x + k(:,2) * aa(3, 2).', args{:});
## compute new time and new values for the unknowns
## t_next = t + dt;
x_next = x + k(:,1:3) * cc(:); # 3rd order approximation
## if the estimation of the error is required
if (nargout >= 3)
## new solution to be compared with the previous one
k(:,4) = feval (fcn, t_next, x_next, args{:});
cc_prime = dt * c_prime;
x_est = x + k * cc_prime(:);
endif
endfunction
|
