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comparison +sbp/+implementations/d2_noneq_variable_6.m @ 1325:1b0f2415237f feature/D2_boundary_opt

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Add variable coefficient boundary-optimized second derivatives.
author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Sun, 13 Feb 2022 19:32:34 +0100
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children 855871e0b852
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1301:8978521b0f06 1325:1b0f2415237f
1 function [H, HI, D1, D2, DI] = d2_noneq_variable_6(N, h, options)
2 % N: Number of grid points
3 % h: grid spacing
4 % options: struct containing options for constructing the operator
5 % current options are:
6 % options.stencil_type ('minimal','nonminimal','wide')
7 % options.AD ('upwind', 'op')
8
9 % BP: Number of boundary points
10 % order: Accuracy of interior stencil
11 BP = 6;
12 order = 6;
13
14 %%%% Norm matrix %%%%%%%%
15 P = zeros(BP, 1);
16 P0 = 1.3030223027124e-01;
17 P1 = 6.8851501587715e-01;
18 P2 = 9.5166202564389e-01;
19 P3 = 9.9103890475697e-01;
20 P4 = 1.0028757074552e+00;
21 P5 = 9.9950151111941e-01;
22
23 for i = 0:BP - 1
24 P(i + 1) = eval(['P' num2str(i)]);
25 end
26
27 Hv = ones(N, 1);
28 Hv(1:BP) = P;
29 Hv(end - BP + 1:end) = flip(P);
30 Hv = h * Hv;
31 H = spdiags(Hv, 0, N, N);
32 HI = spdiags(1 ./ Hv, 0, N, N);
33 %%%%%%%%%%%%%%%%%%%%%%%%%
34
35 %%%% Q matrix %%%%%%%%%%%
36
37 % interior stencil
38 d = [-1/60, 3/20, -3/4, 0, 3/4, -3/20, 1/60];
39 d = repmat(d, N, 1);
40 Q = spdiags(d, -order / 2:order / 2, N, N);
41
42 % Boundaries
43 Q0_0 = -5.0000000000000e-01;
44 Q0_1 = 6.6042071945824e-01;
45 Q0_2 = -2.2104152954203e-01;
46 Q0_3 = 7.6243679810093e-02;
47 Q0_4 = -1.7298206716724e-02;
48 Q0_5 = 1.6753369904210e-03;
49 Q0_6 = 0.0000000000000e+00;
50 Q0_7 = 0.0000000000000e+00;
51 Q0_8 = 0.0000000000000e+00;
52 Q1_0 = -6.6042071945824e-01;
53 Q1_1 = 0.0000000000000e+00;
54 Q1_2 = 8.7352798702787e-01;
55 Q1_3 = -2.6581719253084e-01;
56 Q1_4 = 5.7458484948314e-02;
57 Q1_5 = -4.7485599871040e-03;
58 Q1_6 = 0.0000000000000e+00;
59 Q1_7 = 0.0000000000000e+00;
60 Q1_8 = 0.0000000000000e+00;
61 Q2_0 = 2.2104152954203e-01;
62 Q2_1 = -8.7352798702787e-01;
63 Q2_2 = 0.0000000000000e+00;
64 Q2_3 = 8.1707122038457e-01;
65 Q2_4 = -1.8881125503769e-01;
66 Q2_5 = 2.4226492138960e-02;
67 Q2_6 = 0.0000000000000e+00;
68 Q2_7 = 0.0000000000000e+00;
69 Q2_8 = 0.0000000000000e+00;
70 Q3_0 = -7.6243679810093e-02;
71 Q3_1 = 2.6581719253084e-01;
72 Q3_2 = -8.1707122038457e-01;
73 Q3_3 = 0.0000000000000e+00;
74 Q3_4 = 7.6798636652679e-01;
75 Q3_5 = -1.5715532552963e-01;
76 Q3_6 = 1.6666666666667e-02;
77 Q3_7 = 0.0000000000000e+00;
78 Q3_8 = 0.0000000000000e+00;
79 Q4_0 = 1.7298206716724e-02;
80 Q4_1 = -5.7458484948314e-02;
81 Q4_2 = 1.8881125503769e-01;
82 Q4_3 = -7.6798636652679e-01;
83 Q4_4 = 0.0000000000000e+00;
84 Q4_5 = 7.5266872305402e-01;
85 Q4_6 = -1.5000000000000e-01;
86 Q4_7 = 1.6666666666667e-02;
87 Q4_8 = 0.0000000000000e+00;
88 Q5_0 = -1.6753369904210e-03;
89 Q5_1 = 4.7485599871040e-03;
90 Q5_2 = -2.4226492138960e-02;
91 Q5_3 = 1.5715532552963e-01;
92 Q5_4 = -7.5266872305402e-01;
93 Q5_5 = 0.0000000000000e+00;
94 Q5_6 = 7.5000000000000e-01;
95 Q5_7 = -1.5000000000000e-01;
96 Q5_8 = 1.6666666666667e-02;
97
98 for i = 1:BP
99
100 for j = 1:BP
101 Q(i, j) = eval(['Q' num2str(i - 1) '_' num2str(j - 1)]);
102 Q(N + 1 - i, N + 1 - j) = -eval(['Q' num2str(i - 1) '_' num2str(j - 1)]);
103 end
104
105 end
106
107 %%%%%%%%%%%%%%%%%%%%%%%%%%%
108
109 %%% Undivided difference operators %%%%
110 % Closed with zeros at the first boundary nodes.
111 m = N;
112
113 DD_3 = (-diag(ones(m - 2, 1), -2) + 3 * diag(ones(m - 1, 1), -1) - 3 * diag(ones(m, 1), 0) + diag(ones(m - 1, 1), 1));
114 DD_3(1:5, 1:6) = [0 0 0 0 0 0; 0 0 0 0 0 0; -0.46757024540266021836e1 0.88373748766984018738e1 -0.56477423503490435435e1 0.14860699276772438533e1 0 0; 0 -0.13802450758054908946e1 0.36701915175801340778e1 -0.33643068661005748879e1 0.10743604243259317047e1 0; 0 0 -0.10409288946349185618e1 0.30665535320781497878e1 -0.30329117010471766032e1 0.10072870636039453772e1; ];
115 DD_3(m - 3:m, m - 5:m) = [-0.10072870636039453772e1 0.30329117010471766032e1 -0.30665535320781497878e1 0.10409288946349185618e1 0 0; 0 -0.10743604243259317047e1 0.33643068661005748879e1 -0.36701915175801340778e1 0.13802450758054908946e1 0; 0 0 -0.14860699276772438533e1 0.56477423503490435435e1 -0.88373748766984018738e1 0.46757024540266021836e1; 0 0 0 0 0 0; ];
116 DD_3 = sparse(DD_3);
117
118 DD_4 = (diag(ones(m - 2, 1), 2) - 4 * diag(ones(m - 1, 1), 1) + 6 * diag(ones(m, 1), 0) - 4 * diag(ones(m - 1, 1), -1) + diag(ones(m - 2, 1), -2));
119 DD_4(1:5, 1:7) = [0 0 0 0 0 0 0; 0 0 0 0 0 0 0; 0.57302111593550648941e1 -0.12521994384708052700e2 0.11419402572582197931e2 -0.59442797107089754133e1 0.13166603634797652881e1 0 0; 0 0.14441513881249918393e1 -0.49292485821432017638e1 0.67286137322011497757e1 -0.42974416973037268190e1 0.10539251591207869677e1 0; 0 0 0.10466075357769140419e1 -0.40887380427708663837e1 0.60658234020943532065e1 -0.40291482544157815088e1 0.10054553593153806442e1; ];
120 DD_4(m - 4:m, m - 6:m) = [0.10054553593153806442e1 -0.40291482544157815088e1 0.60658234020943532065e1 -0.40887380427708663837e1 0.10466075357769140419e1 0 0; 0 0.10539251591207869677e1 -0.42974416973037268190e1 0.67286137322011497757e1 -0.49292485821432017638e1 0.14441513881249918393e1 0; 0 0 0.13166603634797652881e1 -0.59442797107089754133e1 0.11419402572582197931e2 -0.12521994384708052700e2 0.57302111593550648941e1; 0 0 0 0 0 0 0; 0 0 0 0 0 0 0; ];
121 DD_4 = sparse(DD_4);
122
123 DD_5 = (-diag(ones(m - 3, 1), -3) + 5 * diag(ones(m - 2, 1), -2) - 10 * diag(ones(m - 1, 1), -1) + 10 * diag(ones(m, 1), 0) - 5 * diag(ones(m - 1, 1), 1) + diag(ones(m - 2, 1), 2));
124 DD_5(1:6, 1:8) = [0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0; -0.67194556014531368457e1 0.16377214352871472626e2 -0.19171027475746103125e2 0.14860699276772438533e2 -0.65833018173988264407e1 0.12358712649541552519e1 0 0; 0 -0.14971527633959360324e1 0.61951742553920293904e1 -0.11214356220335249626e2 0.10743604243259317047e2 -0.52696257956039348385e1 0.10423562806837740594e1 0; 0 0 -0.10511702536596915242e1 0.51109225534635829797e1 -0.10109705670157255344e2 0.10072870636039453772e2 -0.50272767965769032212e1 0.10043595308908133377e1; ];
125 DD_5(m - 4:m, m - 7:m) = [-0.10043595308908133377e1 0.50272767965769032212e1 -0.10072870636039453772e2 0.10109705670157255344e2 -0.51109225534635829797e1 0.10511702536596915242e1 0 0; 0 -0.10423562806837740594e1 0.52696257956039348385e1 -0.10743604243259317047e2 0.11214356220335249626e2 -0.61951742553920293904e1 0.14971527633959360324e1 0; 0 0 -0.12358712649541552519e1 0.65833018173988264407e1 -0.14860699276772438533e2 0.19171027475746103125e2 -0.16377214352871472626e2 0.67194556014531368457e1; 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0; ];
126 DD_5 = sparse(DD_5);
127
128 DD_6 = (diag(ones(m - 3, 1), 3) - 6 * diag(ones(m - 2, 1), 2) + 15 * diag(ones(m - 1, 1), 1) - 20 * diag(ones(m, 1), 0) + 15 * diag(ones(m - 1, 1), -1) - 6 * diag(ones(m - 2, 1), -2) + diag(ones(m - 3, 1), -3));
129 DD_6(1:6, 1:9) = [0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0; 0.76591061528436941127e1 -0.20373923615000091397e2 0.28913418478606999359e2 -0.29721398553544877066e2 0.19749905452196479322e2 -0.74152275897249315116e1 0.11881196746227271813e1 0 0; 0 0.15426631885693469226e1 -0.74666187707188589528e1 0.16821534330502874439e2 -0.21487208486518634095e2 0.15808877386811804515e2 -0.62541376841026443562e1 0.10348900354561115264e1 0; 0 0 0.10549863219420430611e1 -0.61331070641562995756e1 0.15164558505235883016e2 -0.20145741272078907544e2 0.15081830389730709664e2 -0.60261571853448800265e1 0.10036303046714514054e1; ];
130 DD_6(m - 5:m, m - 8:m) = [0.10036303046714514054e1 -0.60261571853448800265e1 0.15081830389730709664e2 -0.20145741272078907544e2 0.15164558505235883016e2 -0.61331070641562995756e1 0.10549863219420430611e1 0 0; 0 0.10348900354561115264e1 -0.62541376841026443562e1 0.15808877386811804515e2 -0.21487208486518634095e2 0.16821534330502874439e2 -0.74666187707188589528e1 0.15426631885693469226e1 0; 0 0 0.11881196746227271813e1 -0.74152275897249315116e1 0.19749905452196479322e2 -0.29721398553544877066e2 0.28913418478606999359e2 -0.20373923615000091397e2 0.76591061528436941127e1; 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0; ];
131 DD_6 = sparse(DD_6);
132
133 %%%% Difference operators %%%
134 D1 = H \ Q;
135
136 % Helper functions for constructing D2(c)
137 % TODO: Consider changing sparse(diag(...)) to spdiags(....)
138
139 % Minimal 7 point stencil width
140 function D2 = D2_fun_minimal(c)
141 % Here we add variable diffusion
142 C1 = sparse(diag(c));
143 C2 = 1/2 * diag(ones(m - 1, 1), -1) + 1/2 * diag(ones(m, 1), 0); C2(1, 2) = 1/2;
144 C3 = 1/3 * diag(ones(m - 1, 1), -1) + 1/3 * diag(ones(m - 1, 1), 1) + 1/3 * diag(ones(m, 1), 0); C3(1, 3) = 1/3; C3(m, m - 2) = 1/3;
145
146 C2 = sparse(diag(C2 * c));
147 C3 = sparse(diag(C3 * c));
148
149 % Remainder term added to wide second derivative operator
150 R = (1/3600 / h) * transpose(DD_6) * C1 * DD_6 + (1/600 / h) * transpose(DD_5) * C2 * DD_5 + (1/80 / h) * transpose(DD_4) * C3 * DD_4;
151 D2 = D1 * C1 * D1 - H \ R;
152 end
153
154 % Few additional grid point in interior stencil cmp. to minimal
155 function D2 = D2_fun_nonminimal(c)
156 % Here we add variable diffusion
157 C1 = sparse(diag(c));
158 C2 = 1/2 * diag(ones(m - 1, 1), -1) + 1/2 * diag(ones(m, 1), 0); C2(1, 2) = 1/2;
159
160 C2 = sparse(diag(C2 * c));
161
162 % Remainder term added to wide second derivative operator
163 R = (1/3600 / h) * transpose(DD_6) * C1 * DD_6 + (1/600 / h) * transpose(DD_5) * C2 * DD_5;
164 D2 = D1 * C1 * D1 - H \ R;
165 end
166
167 % Wide stencil
168 function D2 = D2_fun_wide(c)
169 % Here we add variable diffusion
170 C1 = sparse(diag(c));
171 D2 = D1 * C1 * D1;
172 end
173
174 switch options.stencil_width
175 case 'minimal'
176 D2 = @D2_fun_minimal;
177 case 'nonminimal'
178 D2 = @D2_fun_nonminimal;
179 case 'wide'
180 D2 = @D2_fun_wide;
181 otherwise
182 error('No option %s for stencil width', options.stencil_width)
183 end
184
185 %%%%%%%%%%%%%%%%%%%%%%%%%%%
186
187 %%%% Artificial dissipation operator %%%
188 switch options.AD
189 case 'upwind'
190 % This is the choice that yield 3rd order Upwind
191 DI = H \ (transpose(DD_3) * DD_3) * (-1/60);
192 case 'op'
193 % This choice will preserve the order of the underlying
194 % Non-dissipative D1 SBP operator
195 DI = H \ (transpose(DD_4) * DD_4) * (-1 / (5 * 60));
196 % Notice that you can use any negative number instead of (-1/(5*60))
197 otherwise
198 error("Artificial dissipation options '%s' not implemented.", option.AD)
199 end
200
201 %%%%%%%%%%%%%%%%%%%%%%%%%%%
202 end