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comparison +sbp/+implementations/d4_compatible_halfvariable_2.m @ 261:6009f2712d13 operator_remake

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Moved and renamned all implementations.
author Martin Almquist <martin.almquist@it.uu.se>
date Thu, 08 Sep 2016 15:35:45 +0200
parents
children bfa130b7abf6
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260:b4116ce49ac4 261:6009f2712d13
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 H=diag(ones(m,1),0);H(1,1)=1/2;H(m,m)=1/2;
31
32
33 H=H*h;
34 HI=inv(H);
35
36
37 % First derivative SBP operator, 1st order accurate at first 6 boundary points
38
39 q1=1/2;
40 Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
41
42 %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));
43
44
45 e_1=zeros(m,1);e_1(1)=1;
46 e_m=zeros(m,1);e_m(m)=1;
47
48
49 D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ;
50
51 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
52
53
54
55 % Second derivative, 1st order accurate at first boundary points
56
57 % below for constant coefficients
58 % m1=-1;m0=2;
59 % 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;
60 % M=M/h;
61 %D2=HI*(-M-e_1*S_1+e_m*S_m);
62
63 % Below for variable coefficients
64 % Require a vector c with the koeffients
65
66 S_U=[-3/2 2 -1/2]/h;
67 S_1=zeros(1,m);
68 S_1(1:3)=S_U;
69 S_m=zeros(1,m);
70 S_m(m-2:m)=fliplr(-S_U);
71
72 S_1 = S_1';
73 S_m = S_m';
74
75 M=sparse(m,m);
76 e_1 = sparse(e_1);
77 e_m = sparse(e_m);
78 S_1 = sparse(S_1);
79 S_m = sparse(S_m);
80
81 scheme_width = 3;
82 scheme_radius = (scheme_width-1)/2;
83 r = (1+scheme_radius):(m-scheme_radius);
84
85 function D2 = D2_fun(c)
86
87 Mm1 = -c(r-1)/2 - c(r)/2;
88 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
89 Mp1 = -c(r)/2 - c(r+1)/2;
90
91 M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m);
92
93
94 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;];
95 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;];
96 M=M/h;
97
98 D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
99 end
100 D2 = @D2_fun;
101
102
103
104
105
106 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
107
108
109
110 % Third derivative, 1st order accurate at first 6 boundary points
111
112 q2=1/2;q1=-1;
113 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));
114
115 %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));
116
117
118 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;];
119 Q3(1:4,1:4)=Q3_U;
120 Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) );
121 Q3=Q3/h^2;
122
123
124
125 S2_U=[1 -2 1;]/h^2;
126 S2_1=zeros(1,m);
127 S2_1(1:3)=S2_U;
128 S2_m=zeros(1,m);
129 S2_m(m-2:m)=fliplr(S2_U);
130 S2_1 = S2_1';
131 S2_m = S2_m';
132
133
134
135 D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*S_1*S_1' -1/2*S_m*S_m' ) ;
136
137 % Fourth derivative, 0th order accurate at first 6 boundary points (still
138 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
139
140 m2=1;m1=-4;m0=6;
141 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);
142
143 %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));
144
145 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;];
146
147
148 M4(1:4,1:4)=M4_U;
149
150 M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) );
151 M4=M4/h^3;
152
153 S3_U=[-1 3 -3 1;]/h^3;
154 S3_1=zeros(1,m);
155 S3_1(1:4)=S3_U;
156 S3_m=zeros(1,m);
157 S3_m(m-3:m)=fliplr(-S3_U);
158 S3_1 = S3_1';
159 S3_m = S3_m';
160
161 D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m');
162 end