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comparison +scheme/Wave2d.m @ 0:48b6fb693025

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Initial commit.
author Jonatan Werpers <jonatan@werpers.com>
date Thu, 17 Sep 2015 10:12:50 +0200
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children a8ed986fcf57
comparison
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1 classdef SchmWave2d < noname.Scheme
2 properties
3 m % Number of points in each direction, possibly a vector
4 h % Grid spacing
5 x,y % Grid
6 X,Y % Values of x and y for each grid point
7 order % Order accuracy for the approximation
8
9 D % non-stabalized scheme operator
10 M % Derivative norm
11 alpha
12
13 H % Discrete norm
14 Hi
15 H_x, H_y % Norms in the x and y directions
16 Hx,Hy % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
17 Hi_x, Hi_y
18 Hix, Hiy
19 e_w, e_e, e_s, e_n
20 d1_w, d1_e, d1_s, d1_n
21 gamm_x, gamm_y
22 end
23
24 methods
25 function obj = SchmWave2d(m,xlim,ylim,order,alpha)
26 default_arg('a',1);
27
28 if length(m) == 1
29 m = [m m];
30 end
31
32 m_x = m(1);
33 m_y = m(2);
34
35 [x, h_x] = util.get_grid(xlim{:},m_x);
36 [y, h_y] = util.get_grid(ylim{:},m_y);
37
38 ops_x = sbp.Ordinary(m_x,h_x,order);
39 ops_y = sbp.Ordinary(m_y,h_y,order);
40
41 I_x = speye(m_x);
42 I_y = speye(m_y);
43
44 D2_x = sparse(ops_x.derivatives.D2);
45 H_x = sparse(ops_x.norms.H);
46 Hi_x = sparse(ops_x.norms.HI);
47 M_x = sparse(ops_x.norms.M);
48 e_l_x = sparse(ops_x.boundary.e_1);
49 e_r_x = sparse(ops_x.boundary.e_m);
50 d1_l_x = sparse(ops_x.boundary.S_1);
51 d1_r_x = sparse(ops_x.boundary.S_m);
52
53 D2_y = sparse(ops_y.derivatives.D2);
54 H_y = sparse(ops_y.norms.H);
55 Hi_y = sparse(ops_y.norms.HI);
56 M_y = sparse(ops_y.norms.M);
57 e_l_y = sparse(ops_y.boundary.e_1);
58 e_r_y = sparse(ops_y.boundary.e_m);
59 d1_l_y = sparse(ops_y.boundary.S_1);
60 d1_r_y = sparse(ops_y.boundary.S_m);
61
62 D2 = kr(D2_x, I_y) + kr(I_x, D2_y);
63 obj.H = kr(H_x,H_y);
64 obj.Hx = kr(H_x,I_y);
65 obj.Hy = kr(I_x,H_y);
66 obj.Hix = kr(Hi_x,I_y);
67 obj.Hiy = kr(I_x,Hi_y);
68 obj.Hi = kr(Hi_x,Hi_y);
69 obj.M = kr(M_x,H_y)+kr(H_x,M_y);
70 obj.e_w = kr(e_l_x,I_y);
71 obj.e_e = kr(e_r_x,I_y);
72 obj.e_s = kr(I_x,e_l_y);
73 obj.e_n = kr(I_x,e_r_y);
74 obj.d1_w = kr(d1_l_x,I_y);
75 obj.d1_e = kr(d1_r_x,I_y);
76 obj.d1_s = kr(I_x,d1_l_y);
77 obj.d1_n = kr(I_x,d1_r_y);
78
79 obj.m = m;
80 obj.h = [h_x h_y];
81 obj.order = order;
82
83 obj.alpha = alpha;
84 obj.D = alpha*D2;
85 obj.x = x;
86 obj.y = y;
87 obj.X = kr(x,ones(m_y,1));
88 obj.Y = kr(ones(m_x,1),y);
89
90 obj.gamm_x = h_x*ops_x.borrowing.M.S;
91 obj.gamm_y = h_y*ops_y.borrowing.M.S;
92 end
93
94
95 % Closure functions return the opertors applied to the own doamin to close the boundary
96 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
97 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
98 % type is a string specifying the type of boundary condition if there are several.
99 % data is a function returning the data that should be applied at the boundary.
100 % neighbour_scheme is an instance of Scheme that should be interfaced to.
101 % neighbour_boundary is a string specifying which boundary to interface to.
102 function [closure, penalty] = boundary_condition(obj,boundary,type,data)
103 default_arg('type','neumann');
104 default_arg('data',0);
105
106 [e,d,s,gamm,halfnorm_inv] = obj.get_boundary_ops(boundary);
107
108 switch type
109 % Dirichlet boundary condition
110 case {'D','d','dirichlet'}
111 alpha = obj.alpha;
112
113 % tau1 < -alpha^2/gamma
114 tuning = 1.1;
115 tau1 = -tuning*alpha/gamm;
116 tau2 = s*alpha;
117
118 p = tau1*e + tau2*d;
119
120 closure = halfnorm_inv*p*e';
121
122 pp = halfnorm_inv*p;
123 switch class(data)
124 case 'double'
125 penalty = pp*data;
126 case 'function_handle'
127 penalty = @(t)pp*data(t);
128 otherwise
129 error('Wierd data argument!')
130 end
131
132
133 % Neumann boundary condition
134 case {'N','n','neumann'}
135 alpha = obj.alpha;
136 tau1 = -s*alpha;
137 tau2 = 0;
138 tau = tau1*e + tau2*d;
139
140 closure = halfnorm_inv*tau*d';
141
142 pp = halfnorm_inv*tau;
143 switch class(data)
144 case 'double'
145 penalty = pp*data;
146 case 'function_handle'
147 penalty = @(t)pp*data(t);
148 otherwise
149 error('Wierd data argument!')
150 end
151
152 % Unknown, boundary condition
153 otherwise
154 error('No such boundary condition: type = %s',type);
155 end
156 end
157
158 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
159 % u denotes the solution in the own domain
160 % v denotes the solution in the neighbour domain
161 [e_u,d_u,s_u,gamm_u, halfnorm_inv] = obj.get_boundary_ops(boundary);
162 [e_v,d_v,s_v,gamm_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
163
164 tuning = 1.1;
165
166 alpha_u = obj.alpha;
167 alpha_v = neighbour_scheme.alpha;
168
169 % tau1 < -(alpha_u/gamm_u + alpha_v/gamm_v)
170
171 tau1 = -(alpha_u/gamm_u + alpha_v/gamm_v) * tuning;
172 tau2 = s_u*1/2*alpha_u;
173 sig1 = s_u*(-1/2);
174 sig2 = 0;
175
176 tau = tau1*e_u + tau2*d_u;
177 sig = sig1*e_u + sig2*d_u;
178
179 closure = halfnorm_inv*( tau*e_u' + sig*alpha_u*d_u');
180 penalty = halfnorm_inv*(-tau*e_v' - sig*alpha_v*d_v');
181 end
182
183 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
184 % The right boundary is considered the positive boundary
185 function [e,d,s,gamm, halfnorm_inv] = get_boundary_ops(obj,boundary)
186 switch boundary
187 case 'w'
188 e = obj.e_w;
189 d = obj.d1_w;
190 s = -1;
191 gamm = obj.gamm_x;
192 halfnorm_inv = obj.Hix;
193 case 'e'
194 e = obj.e_e;
195 d = obj.d1_e;
196 s = 1;
197 gamm = obj.gamm_x;
198 halfnorm_inv = obj.Hix;
199 case 's'
200 e = obj.e_s;
201 d = obj.d1_s;
202 s = -1;
203 gamm = obj.gamm_y;
204 halfnorm_inv = obj.Hiy;
205 case 'n'
206 e = obj.e_n;
207 d = obj.d1_n;
208 s = 1;
209 gamm = obj.gamm_y;
210 halfnorm_inv = obj.Hiy;
211 otherwise
212 error('No such boundary: boundary = %s',boundary);
213 end
214 end
215
216 function N = size(obj)
217 N = prod(obj.m);
218 end
219
220 end
221
222 methods(Static)
223 % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u
224 % and bound_v of scheme schm_v.
225 % [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l')
226 function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v)
227 [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v);
228 [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u);
229 end
230 end
231 end