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comparison +sbp/+implementations/d4_compatible_2.m @ 284:dae8c3a56f5e

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Merged in operator_remake (pull request #2) Operator remake
author Jonatan Werpers <jonatan.werpers@it.uu.se>
date Mon, 12 Sep 2016 12:53:02 +0200
parents f7ac3cd6eeaa
children
comparison
equal deleted inserted replaced
282:18c023aaf3f7 284:dae8c3a56f5e
1 function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m,...
2 S3_1, S3_m, S_1, S_m] = d4_compatible_2(m,h)
3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
4 %%% 4:de ordn. SBP Finita differens %%%
5 %%% operatorer framtagna av Ken Mattsson %%%
6 %%% %%%
7 %%% 6 randpunkter, diagonal norm %%%
8 %%% %%%
9 %%% Datum: 2013-11-11 %%%
10 %%% %%%
11 %%% %%%
12 %%% H (Normen) %%%
13 %%% D1 (approx f?rsta derivatan) %%%
14 %%% D2 (approx andra derivatan) %%%
15 %%% D3 (approx tredje derivatan) %%%
16 %%% D2 (approx fj?rde derivatan) %%%
17 %%% %%%
18 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
19
20 % M?ste ange antal punkter (m) och stegl?ngd (h)
21 % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
22 % vi har 3de och 4de derivator i v?r PDE
23 % I annat fall anv?nd de "traditionella" som har noggrannare
24 % randsplutningar f?r D1 och D2
25
26 % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
27 % vilket ?r n?dv?ndigt f?r stabilitet
28
29 BP = 4;
30 if(m<2*BP)
31 error(['Operator requires at least ' num2str(2*BP) ' grid points']);
32 end
33
34 H=speye(m,m);H(1,1)=1/2;H(m,m)=1/2;
35
36
37 H=H*h;
38 HI=inv(H);
39
40
41 % First derivative SBP operator, 1st order accurate at first 6 boundary points
42
43 q1=1/2;
44 % Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
45 stencil = [-q1,0,q1];
46 d = (length(stencil)-1)/2;
47 diags = -d:d;
48 Q = stripeMatrix(stencil, diags, m);
49
50 %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));
51
52
53 e_1=sparse(m,1);e_1(1)=1;
54 e_m=sparse(m,1);e_m(m)=1;
55
56
57 D1=H\(Q-1/2*(e_1*e_1')+1/2*(e_m*e_m')) ;
58
59 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
60
61
62
63 % Second derivative, 1st order accurate at first 6 boundary points
64 % m1=-1;m0=2;
65 % % 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;
66 % stencil = [m2,m1,m0,m1,m2];
67 % d = (length(stencil)-1)/2;
68 % diags = -d:d;
69 % M = stripeMatrix(stencil, diags, m);
70 % M=M/h;
71
72 S_U=[-1 1]/h;
73 S_1=sparse(1,m);
74 S_1(1:2)=S_U;
75 S_m=sparse(1,m);
76
77 S_m(m-1:m)=fliplr(-S_U);
78
79 % D2=H\(-M-e_1*S_1+e_m*S_m);
80
81
82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
83
84
85
86 % Third derivative, 1st order accurate at first 6 boundary points
87
88 % q2=1/2;q1=-1;
89 % % 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));
90 % stencil = [-q2,-q1,0,q1,q2];
91 % d = (length(stencil)-1)/2;
92 % diags = -d:d;
93 % Q3 = stripeMatrix(stencil, diags, m);
94
95 %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));
96
97
98 % Q3_U = [0 -0.2e1 / 0.5e1 0.3e1 / 0.10e2 0.1e1 / 0.10e2; 0.2e1 / 0.5e1 0 -0.7e1 / 0.10e2 0.3e1 / 0.10e2; -0.3e1 / 0.10e2 0.7e1 / 0.10e2 0 -0.9e1 / 0.10e2; -0.1e1 / 0.10e2 -0.3e1 / 0.10e2 0.9e1 / 0.10e2 0;];
99 % Q3(1:4,1:4)=Q3_U;
100 % Q3(m-3:m,m-3:m)=rot90( -Q3_U ,2 );
101 % Q3=Q3/h^2;
102
103
104
105 S2_U=[1 -2 1;]/h^2;
106 S2_1=sparse(1,m);
107 S2_1(1:3)=S2_U;
108 S2_m=sparse(1,m);
109 S2_m(m-2:m)=fliplr(S2_U);
110
111
112
113 % D3=H\(Q3 - e_1*S2_1 + e_m*S2_m +1/2*(S_1'*S_1) -1/2*(S_m'*S_m) ) ;
114
115 % Fourth derivative, 0th order accurate at first 6 boundary points (still
116 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
117
118 m2=1;m1=-4;m0=6;
119 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);
120
121 %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));
122
123 M4_U=[0.4e1 / 0.5e1 -0.7e1 / 0.5e1 0.2e1 / 0.5e1 0.1e1 / 0.5e1; -0.7e1 / 0.5e1 0.16e2 / 0.5e1 -0.11e2 / 0.5e1 0.2e1 / 0.5e1; 0.2e1 / 0.5e1 -0.11e2 / 0.5e1 0.21e2 / 0.5e1 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;];
124
125 M4(1:4,1:4)=M4_U;
126
127 M4(m-3:m,m-3:m)=rot90( M4_U ,2 );
128 M4=M4/h^3;
129
130 S3_U=[-1 3 -3 1;]/h^3;
131 S3_1=sparse(1,m);
132 S3_1(1:4)=S3_U;
133 S3_m=sparse(1,m);
134 S3_m(m-3:m)=fliplr(-S3_U);
135
136 D4=H\(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m);
137
138
139
140 S_1 = S_1';
141 S_m = S_m';
142 S2_1 = S2_1';
143 S2_m = S2_m';
144 S3_1 = S3_1';
145 S3_m = S3_m';
146
147
148
149
150 % L=h*(m-1);
151
152 % x1=linspace(0,L,m)';
153 % x2=x1.^2/fac(2);
154 % x3=x1.^3/fac(3);
155 % x4=x1.^4/fac(4);
156 % x5=x1.^5/fac(5);
157
158 % x0=x1.^0/fac(1);
159
160
161 end