-
Notifications
You must be signed in to change notification settings - Fork 0
/
test_problem_3.m
186 lines (118 loc) · 4.47 KB
/
test_problem_3.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
183
184
185
186
%test_problem_3.m written 3-30-16 by JTN to run test problem 1 of Thackham
%2009 work -- simulate 2 D diffusion-convection problem
%Does not work for n even if x0,y0 = 0.5 (ratiomal?) .... why???
clear all; clc
%initialization, parameters
n = 75;
dt = 1e-3;
x = linspace(0,1,n);
y = linspace(0,1,n);
[X,Y] = meshgrid(x,y);
dx = x(2) - x(1);
dy = y(2) - y(1);
t = 0:dt:1;
xn = length(x);
yn = length(y);
tn = length(t);
Dx = 1e-2;
Dy = 1e-2;
chix = 8;
chiy = 8;
Dx_c = Dx*dt/dx^2;
Dy_c = Dy*dt/dy^2;
Vx_c = chix*dt/dx;
Vy_c = chiy*dt/dy;
x0 = 0.5;
y0 = 0.5;
theta = 0.5;
%specify boundary nodes
xbd_0 = 1:yn;
xbd_1 = yn*(xn-1)+1:xn*yn;
xbd = union(xbd_0,xbd_1);
ybd_0 = 1:xn:yn*(xn-1)+1;
ybd_1 = yn:yn:xn*yn;
ybd = union(ybd_0,ybd_1);
bd = union(xbd,ybd);
%indices for r_s in computation.
y_ind_1 = 1:yn-2:(xn-2)*(yn-2);
y_ind_nm1 = yn-2:yn-2:(xn-2)*(yn-2);
%specify interior points
xy_int = 1:xn*yn;
xy_int(bd) = [];
exact_soln = @(x,y,t) (1/(4*t+1))*exp(-(x-chix*t-x0).^2/(Dx*(4*t+1))-(y-chiy*t-y0).^2/(Dy*(4*t+1)));
%initial condition
IC = @(x,y) exact_soln(x,y,0);
%boundary conditions
BC_x_0 = @(y,t) exact_soln(0,y,t);
BC_x_1 = @(y,t) exact_soln(1,y,t);
BC_y_0 = @(x,t) exact_soln(x,0,t);
BC_y_1 = @(x,t) exact_soln(x,1,t);
%sigma for flux limiters
sigma = @(r) (r+abs(r))./(1+abs(r));
%initialize
u = zeros(yn*xn,tn);
u0 = IC(X,Y);
u(:,1) = u0(:);
%sparse matrix as a function for computation
A_pos = @(se,sw,D,v,ind,dn) sparse([ind ind ind],[ind-dn ind ind+dn],[(-D+-v+v.*sw/2); ...
(2*D+v-v.*se/2-v.*sw/2); (-D+v.*se/2)],xn*yn,xn*yn);
A_neg = @(se,sw,D,v,ind,dn) sparse([ind ind ind],[ind-dn ind ind+dn],[(-D-v.*sw/2); ...
(2*D+v.*se/2+v.*sw/2-v); (-D+v-v.*se/2)],xn*yn,xn*yn);
tic
for i = 2:tn
%set BC
u(xbd_0,i) = BC_x_0(y,t(i));
u(xbd_1,i) = BC_x_1(y,t(i));
u(ybd_0,i) = BC_y_0(x,t(i));
u(ybd_1,i) = BC_y_1(x,t(i));
%get Ax matrix
if chix >= 0
r_e = (u(xy_int,i-1) - u(xy_int-yn,i-1))./(u(xy_int+yn,i-1) - u(xy_int,i-1));
%for r_w, start at yn-1 to avoid sampling off the grid ... assign
%value of -1 for first values (would be true for 0 Neumann BC)
r_w = (u(xy_int(yn-1:end)-yn,i-1) - u(xy_int(yn-1:end)-2*yn,i-1))./(u(xy_int(yn-1:end),i-1) - u(xy_int(yn-1:end)-yn,i-1));
r_w = [-1*ones(yn-2,1);r_w];
Ax = A_pos(sigma(r_e),sigma(r_w),Dx_c,Vx_c,xy_int,yn);
elseif chix <0
%remove last x strip to avoid sampling off grid ... replace with -1
r_e = (u(xy_int(1:end-yn+2)+yn,i-1) - u(xy_int(1:end-yn+2)+2*yn,i-1))./(u(xy_int(1:end-yn+2),i-1) - u(xy_int(1:end-yn+2)+yn,i-1));
r_e = [r_e;-1*ones(yn-2,1)];
r_w = (u(xy_int,i-1) - u(xy_int+yn,i-1))./(u(xy_int-yn,i-1) - u(xy_int,i-1));
Ax = A_neg(sigma(r_e),sigma(r_w),Dx_c,Vx_c,xy_int,yn);
end
%get Ay matrix
%get Ax matrix
if chiy >= 0
r_n = (u(xy_int,i-1) - u(xy_int-1,i-1))./(u(xy_int+1,i-1) - u(xy_int,i-1));
r_s = (u(xy_int-1,i-1) - u(xy_int-2,i-1))./(u(xy_int,i-1) - u(xy_int-1,i-1));
%for r_s, note that the row corresponding to y=deltay is sampling
%incorrectly with the second upwind point -- set these sensors = -1
%(would be true with 0 Neumann BC)
r_s(y_ind_1) = -1;
Ay = A_pos(sigma(r_n),sigma(r_s),Dy_c,Vy_c,xy_int,1);
elseif chiy < 0
r_n = (u(xy_int+1,i-1) - u(xy_int+2,i-1))./(u(xy_int,i-1) - u(xy_int+1,i-1));
%for r_n, note that the row corresponding to y=deltay is sampling
%incorrectly with the second upwind point -- set these sensors = -1
%(would be true with 0 Neumann BC)
r_n(y_ind_nm1) = -1;
r_s = (u(xy_int,i-1) - u(xy_int+1,i-1))./(u(xy_int-1,i-1) - u(xy_int,i-1));
Ay = A_neg(sigma(r_n),sigma(r_s),Dy_c,Vy_c,xy_int,1);
end
% (I + theta*(Ap + An))u(:,i) = (I-(1-theta)*(Ap + An))*u(:,i-1)
% u(:,i) = (speye(xn*yn) + theta*(Ax + Ay))\(speye(xn*yn) - (1-theta)*(Ax + Ay))*u(:,i-1);
[u(:,i),flag] = gmres((speye(xn*yn) + theta*(Ax + Ay)),(speye(xn*yn) - (1-theta)*(Ax + Ay))*u(:,i-1));
end
toc
%
for i = 1:10:tn
subplot(1,2,1)
contourf(x,y,reshape(u(:,i),yn,xn),'edgecolor','none')
title('sim')
view(2)
subplot(1,2,2)
contourf(x,y,exact_soln(X,Y,t(i)),'edgecolor','none')
title('exact')
view(2)
pause(.125)
end