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comparison +sbp/+implementations/d4_variable_2.m @ 310:ffa5d557942b feature/beams

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Moved operator implementations and fixed some naming.
author Jonatan Werpers <jonatan@werpers.com>
date Fri, 23 Sep 2016 14:55:08 +0200
parents +sbp/+implementations/d4_compatible_halfvariable_2.m@f7ac3cd6eeaa
children 713b125038a3
comparison
equal deleted inserted replaced
309:8fafe97bf27b 310:ffa5d557942b
1 % Returns D2 as a function handle
2 function [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1,...
3 S2_m, S3_1, S3_m, S_1, S_m] = d4_compatible_halfvariable_2(m,h)
4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
5 %%% 4:de ordn. SBP Finita differens %%%
6 %%% operatorer framtagna av Ken Mattsson %%%
7 %%% %%%
8 %%% 6 randpunkter, diagonal norm %%%
9 %%% %%%
10 %%% Datum: 2013-11-11 %%%
11 %%% %%%
12 %%% %%%
13 %%% H (Normen) %%%
14 %%% D1 (approx f?rsta derivatan) %%%
15 %%% D2 (approx andra derivatan) %%%
16 %%% D3 (approx tredje derivatan) %%%
17 %%% D2 (approx fj?rde derivatan) %%%
18 %%% %%%
19 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
20
21 % M?ste ange antal punkter (m) och stegl?ngd (h)
22 % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
23 % vi har 3de och 4de derivator i v?r PDE
24 % I annat fall anv?nd de "traditionella" som har noggrannare
25 % randsplutningar f?r D1 och D2
26
27 % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
28 % vilket ?r n?dv?ndigt f?r stabilitet
29
30 BP = 4;
31 if(m<2*BP)
32 error(['Operator requires at least ' num2str(2*BP) ' grid points']);
33 end
34
35 H=speye(m,m);H(1,1)=1/2;H(m,m)=1/2;
36
37
38 H=H*h;
39 HI=inv(H);
40
41
42 % First derivative SBP operator, 1st order accurate at first 6 boundary points
43
44 q1=1/2;
45 % Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
46 stencil = [-q1,0,q1];
47 d = (length(stencil)-1)/2;
48 diags = -d:d;
49 Q = stripeMatrix(stencil, diags, m);
50
51 %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2));
52
53
54 e_1=sparse(m,1);e_1(1)=1;
55 e_m=sparse(m,1);e_m(m)=1;
56
57
58 D1=HI*(Q-1/2*(e_1*e_1')+1/2*(e_m*e_m')) ;
59
60 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
61
62
63
64 % Second derivative, 1st order accurate at first boundary points
65
66 % below for constant coefficients
67 % m1=-1;m0=2;
68 % M=m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);M(1,1)=1;M(m,m)=1;
69 % M=M/h;
70 %D2=HI*(-M-e_1*S_1+e_m*S_m);
71
72 % Below for variable coefficients
73 % Require a vector c with the koeffients
74
75 S_U=[-3/2 2 -1/2]/h;
76 S_1=sparse(1,m);
77 S_1(1:3)=S_U;
78 S_m=sparse(1,m);
79 S_m(m-2:m)=fliplr(-S_U);
80
81 S_1 = S_1';
82 S_m = S_m';
83
84 M=sparse(m,m);
85 e_1 = sparse(e_1);
86 e_m = sparse(e_m);
87 S_1 = sparse(S_1);
88 S_m = sparse(S_m);
89
90 scheme_width = 3;
91 scheme_radius = (scheme_width-1)/2;
92 r = (1+scheme_radius):(m-scheme_radius);
93
94 function D2 = D2_fun(c)
95
96 Mm1 = -c(r-1)/2 - c(r)/2;
97 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
98 Mp1 = -c(r)/2 - c(r+1)/2;
99
100 M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m);
101
102
103 M(1:2,1:2)=[c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;];
104 M(m-1:m,m-1:m)=[c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;];
105 M=M/h;
106
107 D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
108 end
109 D2 = @D2_fun;
110
111
112
113
114
115 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
116
117
118
119 % Third derivative, 1st order accurate at first 6 boundary points
120
121 q2=1/2;q1=-1;
122 % Q3=q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
123 stencil = [-q2,-q1,0,q1,q2];
124 d = (length(stencil)-1)/2;
125 diags = -d:d;
126 Q3 = stripeMatrix(stencil, diags, m);
127
128 %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3));
129
130
131 Q3_U = [0 -0.13e2 / 0.16e2 0.7e1 / 0.8e1 -0.1e1 / 0.16e2; 0.13e2 / 0.16e2 0 -0.23e2 / 0.16e2 0.5e1 / 0.8e1; -0.7e1 / 0.8e1 0.23e2 / 0.16e2 0 -0.17e2 / 0.16e2; 0.1e1 / 0.16e2 -0.5e1 / 0.8e1 0.17e2 / 0.16e2 0;];
132 Q3(1:4,1:4)=Q3_U;
133 Q3(m-3:m,m-3:m)=rot90( -Q3_U ,2 );
134 Q3=Q3/h^2;
135
136
137
138 S2_U=[1 -2 1;]/h^2;
139 S2_1=sparse(1,m);
140 S2_1(1:3)=S2_U;
141 S2_m=sparse(1,m);
142 S2_m(m-2:m)=fliplr(S2_U);
143 S2_1 = S2_1';
144 S2_m = S2_m';
145
146
147
148 D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*(S_1*S_1') -1/2*(S_m*S_m') ) ;
149
150 % Fourth derivative, 0th order accurate at first 6 boundary points (still
151 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
152
153 m2=1;m1=-4;m0=6;
154 % M4=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);
155 stencil = [m2,m1,m0,m1,m2];
156 d = (length(stencil)-1)/2;
157 diags = -d:d;
158 M4 = stripeMatrix(stencil, diags, m);
159
160 %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0));
161
162 M4_U=[0.13e2 / 0.10e2 -0.12e2 / 0.5e1 0.9e1 / 0.10e2 0.1e1 / 0.5e1; -0.12e2 / 0.5e1 0.26e2 / 0.5e1 -0.16e2 / 0.5e1 0.2e1 / 0.5e1; 0.9e1 / 0.10e2 -0.16e2 / 0.5e1 0.47e2 / 0.10e2 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;];
163
164
165 M4(1:4,1:4)=M4_U;
166
167 M4(m-3:m,m-3:m)=rot90( M4_U ,2 );
168 M4=M4/h^3;
169
170 S3_U=[-1 3 -3 1;]/h^3;
171 S3_1=sparse(1,m);
172 S3_1(1:4)=S3_U;
173 S3_m=sparse(1,m);
174 S3_m(m-3:m)=fliplr(-S3_U);
175 S3_1 = S3_1';
176 S3_m = S3_m';
177
178 D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m');
179 end