-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathSDIRK_general_corrected_clipped.m
182 lines (141 loc) · 4.48 KB
/
SDIRK_general_corrected_clipped.m
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
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
%%
% t0 - start time
% tf - end time
% h - step size
% y0 - initial value
% ode_func - ODE function function handle
% sdirk_method - function that returns constant coefficients associated with method
% jacobian - Jacobian function
%%
function [t,y] = SDIRK_general_corrected_clipped(t0, tf, h, y0, ode_func, sdirk_method, matrix_func, jacobian )
% Create time grid
Nsteps = ceil( (tf-t0) / h );
t=linspace(t0,tf,Nsteps+1);
% Initialize solution matrix
y = zeros(length(y0),Nsteps+1);
y(:,1) = y0;
% Coefficients for SDIRK method
[A, b, gamma] = SDIRK_coefficients(sdirk_method);
c = sum(A, 2);
stiffaccurate = all(b == A(end,:));
epsilon = 1.e-3;
% Number of stages
s = length(b);
% Initialize stages
F = zeros(length(y0), s);
Y = zeros(length(y0), s);
opt.Display = 'off';
opt.StepTolerance = 1e-8;
opt.FunctionTolerance = 1e-8;
% Loop over time
for i = 2:length(t)
dt = t(i) - t(i - 1);
% Loop over stages
for istage = 1:s
U = y(:,i-1);
for j = 1:(istage - 1)
U = U + dt*A(istage, j)*F(:,j);
end
func_y = @(ys) U - ys + dt*gamma*ode_func(t(i-1) + dt*c(istage), ys);
y_stage = newton_iteration(func_y, @(Y)-eye(length(U))+ dt*gamma*jacobian(t(i-1)+dt*c(istage), Y), y(:,i-1));
Y(:,istage) = max(y_stage,0);
F(:,istage) = ode_func(t(i-1) + dt*c(istage), Y(:,istage));
end
% Update solution
if ~stiffaccurate
y_new = y(:,i-1) + dt*( F(:,:)*b(:) );
else
y_new = y_stage;
end
y_new = max(y_new, epsilon);
Fmean = zeros(length(y0));
for idx = 1:s
Fmean = Fmean + b(idx)*matrix_func(Y(:,idx), t(i-1) + dt*c(idx))*diag(Y(:,idx)./y_new);
end
y(:,i) = (eye(length(Fmean)) - dt*Fmean)\y(:,i-1);
end
end
%%
function u_f = newton_iteration(func, jac, u_0)
i = 0;
while i < 100
c_i = -jac(u_0) \ func(u_0);
u_f = c_i + u_0;
i = i + 1;
if norm(c_i) < 1.e-12
return;
end
u_0 = u_f;
end
end
function [A, b, gamma] = SDIRK_coefficients(method)
switch (method)
case(1)
% 2 stages
gamma = .2928932188134524755991556378951510d0;
A(1,1) = .2928932188134524755991556378951510d0;
A(2,1) = .7071067811865475244008443621048490d0;
A(2,2) = .2928932188134524755991556378951510d0;
b(1) = .7071067811865475244008443621048490d0;
b(2) = .2928932188134524755991556378951510d0;
case(2)
% 2 stages
x = 1 - sqrt(2)/2;
gamma = x;
A = [[x,0];[1-x,x]];
b = A(end,:);
case(3)
% 3 stages
gamma = 1/3;
A = [[1/3,0,0];[1/6,1/3,0];[5/6,-5/12,1/3]];
b = [6/5, -1, 4/5];
case(4)
% 5 stages
gamma = 1/4;
A = [[1/4, 0, 0, 0, 0];...
[13/20, 1/4, 0, 0, 0];...
[580/1287, -175/5148, 1/4, 0, 0];...
[12698/37375, -201/2990, 891/11500, 1/4, 0];...
[944/1365, -400/819, 99/35, -575/252, 1/4]];
b = A(end,:);
case(5)
% 5 stages
gamma = .25d0;
A(1,1) = 0.25d0;
A(2,1) = 0.5d00;
A(2,2) = 0.25d0;
A(3,1) = 0.34d0;
A(3,2) =-0.40d-1;
A(3,3) = 0.25d0;
A(4,1) = 0.2727941176470588235294117647058824d0;
A(4,2) =-0.5036764705882352941176470588235294d-1;
A(4,3) = 0.2757352941176470588235294117647059d-1;
A(4,4) = 0.25d0;
A(5,1) = 1.041666666666666666666666666666667d0;
A(5,2) =-1.020833333333333333333333333333333d0;
A(5,3) = 7.812500000000000000000000000000000d0;
A(5,4) =-7.083333333333333333333333333333333d0;
A(5,5) = 0.25d0;
b(1) = 1.041666666666666666666666666666667d0;
b(2) = -1.020833333333333333333333333333333d0;
b(3) = 7.812500000000000000000000000000000d0;
b(4) = -7.083333333333333333333333333333333d0;
b(5) = 0.250000000000000000000000000000000d0;
case(6)
gamma = 1/4;
A = [[1/4, 0, 0, 0, 0];...
[1/2, 1/4, 0, 0, 0];...
[17/50, -1/25, 1/4, 0, 0];...
[371/1360, -137/2720, 15/544, 1/4, 0];...
[25/24, -49/48, 125/16, -85/12, 1/4]];
b = A(end,:);
otherwise
% 2 stages
gamma = .2928932188134524755991556378951510d0;
A(1,1) = .2928932188134524755991556378951510d0;
A(2,1) = .7071067811865475244008443621048490d0;
A(2,2) = .2928932188134524755991556378951510d0;
b(1) = .7071067811865475244008443621048490d0;
b(2) = .2928932188134524755991556378951510d0;
end
end