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
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
|
#include "headcpp.h"
#include "math.h"
#include "tbl.h"
#include "matrice.h"
#include "leg.h"
double f_left (double) {
return 1. ;
}
double f_right (double) {
return 0. ;
}
double f_source (double x) {
if (x<0)
return 1. ;
else
if (x>0) return 0. ;
else
return 0.5 ;
}
double f_solution (double x) {
double ee = exp(1.) ;
static double bmoins = -1./8./(1+ee*ee) - ee*ee/8./(1+ee*ee*ee*ee) ;
static double bplus = ee*ee*ee*ee/8.*(ee*ee/(1+ee*ee*ee*ee)-1./(1+ee*ee)) ;
if (x<0)
return 0.25 -(ee*ee/4.+bmoins*ee*ee*ee*ee)*exp(2*x)
+ bmoins*exp(-2*x) ;
else
return bplus*(exp(-2*x)-exp(2*x)/ee/ee/ee/ee) ;
}
//*******************
// First derivative
//*******************
Tbl ope_der (const Tbl& so) {
// Verification of size :
assert (so.get_ndim() == 1) ;
int size = so.get_dim(0) ;
// The result is set to zero
Tbl res (size) ;
res.annule_hard() ;
// the computation
for (int n=0 ; n<size ; n++) {
for (int p=n+1 ; p<size ; p++)
if ((p+n)%2 == 1)
res.set(n) += so(p) ;
res.set(n) *= (2*n+1) ;
}
return res ;
}
int main () {
int nr ;
cout << "Please enter the number of coefficients :" << endl ;
cin >> nr ;
// Collocation and weights
Tbl coloc (nr) ;
coloc.set_etat_qcq() ;
Tbl poids (nr) ;
poids.set_etat_qcq() ;
coloc_poids_leg(nr, coloc, poids) ;
// Computation of the matrix D :
Matrice ope_D (nr, nr) ;
ope_D.set_etat_qcq() ;
Tbl val (nr) ;
for (int i=0 ; i<nr ; i++) {
val.annule_hard() ;
val.set(i) = 1 ;
Tbl coef (coef_leg(val)) ;
Tbl der (ope_der(coef)) ;
// Value of derivative at point x_j :
for (int j=0 ; j<nr ; j++) {
double sum = 0 ;
for (int k=0 ; k<nr ; k++)
sum += der(k)*leg(k, coloc(j)) ;
ope_D.set(j,i) = 2*sum ;
}
}
cout << "Derivative operator D : " << endl ;
cout << ope_D << endl ;
// Matrice of the system :
Matrice systeme(2*nr-1, 2*nr-1) ;
systeme.annule_hard() ;
// Left boundary condition :
systeme.set(0,0) = 1 ;
// Equation inside left domain and at the interface:
for (int n=1 ; n<nr ; n++) {
for (int j=0 ; j<nr ; j++)
for (int i=0 ; i<nr ; i++)
systeme.set(n,j) += ope_D(i,j)*ope_D(i,n)*poids(i) ;
systeme.set(n,n) += 4*poids(n) ;
}
// equation at the interface and inside domain 1 :
for (int n=0 ; n<nr-1 ; n++) {
for (int j=0 ; j<nr ; j++)
for (int i=0 ; i<nr ; i++)
systeme.set(n+nr-1,j+nr-1) += ope_D(i,j)*ope_D(i,n)*poids(i) ;
systeme.set(n+nr-1,n+nr-1) += 4*poids(n) ;
}
// Boundary on the right :
systeme.set(2*nr-2, 2*nr-2) = 1 ;
cout << "Matrix of the system: " << endl ;
cout << systeme << endl ;
systeme.set_lu() ;
// Construction of the second member :
Tbl sec_member (2*nr-1) ;
sec_member.annule_hard() ;
// Left boundary condition :
sec_member.set(0) = 0 ;
// domain 1 ;
for (int i=1 ; i<nr ; i++)
sec_member.set(i) = f_left(coloc(i))*poids(i) ;
// domaine 2 ;
for (int i=0 ; i<nr-1 ; i++)
sec_member.set(i+nr-1) += f_right(coloc(i))*poids(i) ;
// right boundary condition :
sec_member.set(2*nr-2) = 0 ;
cout << "Second member of the system" << endl ;
cout << sec_member << endl ;
// Inversion of the system :
Tbl inv (systeme.inverse(sec_member)) ;
// Solution in both domain, at collocations points :
Tbl sol_left (nr) ;
sol_left.set_etat_qcq() ;
for (int i=0 ; i<nr ; i++)
sol_left.set(i) = inv(i) ;
Tbl sol_right (nr) ;
sol_right.set_etat_qcq() ;
for (int i=0 ; i<nr ; i++)
sol_right.set(i) = inv(nr-1+i) ;
cout << "Solutions in both domains, at collocation points : "<< endl ;
cout << sol_left << endl ;
cout << sol_right << endl ;
// Compute the coefficients :
Tbl coef_left (coef_leg(sol_left)) ;
Tbl coef_right (coef_leg(sol_right)) ;
// Outputs :
// Output in a file for plotting purposes :
char name_out[30] ;
sprintf(name_out, "plot_variational_%i.dat", nr) ;
ofstream fiche (name_out) ;
int resolution = 200 ;
double x=-1 ;
double step = 2./resolution ;
// We will also compute the maximum difference between the solution and its numerical value :
double error_max = 0 ;
double xi ;
for (int i=0 ; i<resolution+1 ; i++) {
// in which domain ?
if (x<0)
xi = 2*x+1 ;
else
xi = 2*x-1 ;
// One computes the solution at the current point
double val_func = 0 ;
for (int j=0 ; j<nr ; j++)
if (x<0)
val_func += coef_left(j)*leg(j, xi) ;
else
val_func += coef_right(j)*leg(j, xi) ;
fiche << x << " " << val_func << " " << f_solution(x) << endl ;
double error = fabs(val_func-f_solution(x)) ;
if (error > error_max)
error_max = error ;
x += step ;
}
cout << "Error max : " << endl ;
cout << error_max << endl ;
return 0 ;
}
|
