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annotate +sbp/+implementations/d2_noneq_variable_6.m @ 1332:8e9df030a0a5 feature/D2_boundary_opt

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author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Wed, 04 May 2022 08:42:13 +0200
parents 855871e0b852
children bcdb14b05d03
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1325
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
1 function [H, HI, D1, D2, DI] = d2_noneq_variable_6(N, h, options)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
2 % N: Number of grid points
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
3 % h: grid spacing
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
4 % options: struct containing options for constructing the operator
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
5 % current options are:
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
6 % options.stencil_type ('minimal','nonminimal','wide')
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
7 % options.AD ('upwind', 'op')
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
8
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
9 % BP: Number of boundary points
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
10 % order: Accuracy of interior stencil
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
11 BP = 6;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
12 order = 6;
1330
855871e0b852 Add size checks to the operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1325
diff changeset
13 if(N<2*BP)
855871e0b852 Add size checks to the operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1325
diff changeset
14 error(['Operator requires at least ' num2str(2*BP) ' grid points']);
855871e0b852 Add size checks to the operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1325
diff changeset
15 end
1325
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
16
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
17 %%%% Norm matrix %%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
18 P = zeros(BP, 1);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
19 P0 = 1.3030223027124e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
20 P1 = 6.8851501587715e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
21 P2 = 9.5166202564389e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
22 P3 = 9.9103890475697e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
23 P4 = 1.0028757074552e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
24 P5 = 9.9950151111941e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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25
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
26 for i = 0:BP - 1
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
27 P(i + 1) = eval(['P' num2str(i)]);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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28 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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29
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
30 Hv = ones(N, 1);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
31 Hv(1:BP) = P;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
32 Hv(end - BP + 1:end) = flip(P);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
33 Hv = h * Hv;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
34 H = spdiags(Hv, 0, N, N);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
35 HI = spdiags(1 ./ Hv, 0, N, N);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
36 %%%%%%%%%%%%%%%%%%%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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37
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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38 %%%% Q matrix %%%%%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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39
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
40 % interior stencil
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
41 d = [-1/60, 3/20, -3/4, 0, 3/4, -3/20, 1/60];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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42 d = repmat(d, N, 1);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
43 Q = spdiags(d, -order / 2:order / 2, N, N);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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44
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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45 % Boundaries
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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46 Q0_0 = -5.0000000000000e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
47 Q0_1 = 6.6042071945824e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
48 Q0_2 = -2.2104152954203e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
49 Q0_3 = 7.6243679810093e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
50 Q0_4 = -1.7298206716724e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
51 Q0_5 = 1.6753369904210e-03;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
52 Q0_6 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
53 Q0_7 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
54 Q0_8 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
55 Q1_0 = -6.6042071945824e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
56 Q1_1 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
57 Q1_2 = 8.7352798702787e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
58 Q1_3 = -2.6581719253084e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
59 Q1_4 = 5.7458484948314e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
60 Q1_5 = -4.7485599871040e-03;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
61 Q1_6 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
62 Q1_7 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
63 Q1_8 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
64 Q2_0 = 2.2104152954203e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
65 Q2_1 = -8.7352798702787e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
66 Q2_2 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
67 Q2_3 = 8.1707122038457e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
68 Q2_4 = -1.8881125503769e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
69 Q2_5 = 2.4226492138960e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
70 Q2_6 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
71 Q2_7 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
72 Q2_8 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
73 Q3_0 = -7.6243679810093e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
74 Q3_1 = 2.6581719253084e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
75 Q3_2 = -8.1707122038457e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
76 Q3_3 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
77 Q3_4 = 7.6798636652679e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
78 Q3_5 = -1.5715532552963e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
79 Q3_6 = 1.6666666666667e-02;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
80 Q3_7 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
81 Q3_8 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
82 Q4_0 = 1.7298206716724e-02;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
83 Q4_1 = -5.7458484948314e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
84 Q4_2 = 1.8881125503769e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
85 Q4_3 = -7.6798636652679e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
86 Q4_4 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
87 Q4_5 = 7.5266872305402e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
88 Q4_6 = -1.5000000000000e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
89 Q4_7 = 1.6666666666667e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
90 Q4_8 = 0.0000000000000e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
91 Q5_0 = -1.6753369904210e-03;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
92 Q5_1 = 4.7485599871040e-03;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
93 Q5_2 = -2.4226492138960e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
94 Q5_3 = 1.5715532552963e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
95 Q5_4 = -7.5266872305402e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
96 Q5_5 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
97 Q5_6 = 7.5000000000000e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
98 Q5_7 = -1.5000000000000e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
99 Q5_8 = 1.6666666666667e-02;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
100
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
101 for i = 1:BP
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
102
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
103 for j = 1:BP
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
104 Q(i, j) = eval(['Q' num2str(i - 1) '_' num2str(j - 1)]);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
105 Q(N + 1 - i, N + 1 - j) = -eval(['Q' num2str(i - 1) '_' num2str(j - 1)]);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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106 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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107
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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108 end
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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109
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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110 %%%%%%%%%%%%%%%%%%%%%%%%%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
111
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
112 %%% Undivided difference operators %%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
113 % Closed with zeros at the first boundary nodes.
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
114 m = N;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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115
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
116 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));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
117 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; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
118 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
119 DD_3 = sparse(DD_3);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
120
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
121 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));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
122 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
123 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
124 DD_4 = sparse(DD_4);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
125
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
126 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));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
127 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
128 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
129 DD_5 = sparse(DD_5);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
130
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
131 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));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
132 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
133 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; ];
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
134 DD_6 = sparse(DD_6);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
135
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
136 %%%% Difference operators %%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
137 D1 = H \ Q;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
138
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
139 % Helper functions for constructing D2(c)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
140 % TODO: Consider changing sparse(diag(...)) to spdiags(....)
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
141
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
142 % Minimal 7 point stencil width
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
143 function D2 = D2_fun_minimal(c)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
144 % Here we add variable diffusion
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
145 C1 = sparse(diag(c));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
146 C2 = 1/2 * diag(ones(m - 1, 1), -1) + 1/2 * diag(ones(m, 1), 0); C2(1, 2) = 1/2;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
147 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;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
148
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
149 C2 = sparse(diag(C2 * c));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
150 C3 = sparse(diag(C3 * c));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
151
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
152 % Remainder term added to wide second derivative operator
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
153 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;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
154 D2 = D1 * C1 * D1 - H \ R;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
155 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
156
1332
8e9df030a0a5 Clarify comments
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1330
diff changeset
157 % Non-minimal 9 point stencil width
1325
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
158 function D2 = D2_fun_nonminimal(c)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
159 % Here we add variable diffusion
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
160 C1 = sparse(diag(c));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
161 C2 = 1/2 * diag(ones(m - 1, 1), -1) + 1/2 * diag(ones(m, 1), 0); C2(1, 2) = 1/2;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
162
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
163 C2 = sparse(diag(C2 * c));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
164
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
165 % Remainder term added to wide second derivative operator
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
166 R = (1/3600 / h) * transpose(DD_6) * C1 * DD_6 + (1/600 / h) * transpose(DD_5) * C2 * DD_5;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
167 D2 = D1 * C1 * D1 - H \ R;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
168 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
169
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
170 % Wide stencil
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
171 function D2 = D2_fun_wide(c)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
172 % Here we add variable diffusion
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
173 C1 = sparse(diag(c));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
174 D2 = D1 * C1 * D1;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
175 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
176
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
177 switch options.stencil_width
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
178 case 'minimal'
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
179 D2 = @D2_fun_minimal;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
180 case 'nonminimal'
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
181 D2 = @D2_fun_nonminimal;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
182 case 'wide'
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
183 D2 = @D2_fun_wide;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
184 otherwise
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
185 error('No option %s for stencil width', options.stencil_width)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
186 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
187
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
188 %%%%%%%%%%%%%%%%%%%%%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
189
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
190 %%%% Artificial dissipation operator %%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
191 switch options.AD
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
192 case 'upwind'
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
193 % This is the choice that yield 3rd order Upwind
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
194 DI = H \ (transpose(DD_3) * DD_3) * (-1/60);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
195 case 'op'
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
196 % This choice will preserve the order of the underlying
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
197 % Non-dissipative D1 SBP operator
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
198 DI = H \ (transpose(DD_4) * DD_4) * (-1 / (5 * 60));
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
199 % Notice that you can use any negative number instead of (-1/(5*60))
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
200 otherwise
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
201 error("Artificial dissipation options '%s' not implemented.", option.AD)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
202 end
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
203
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
204 %%%%%%%%%%%%%%%%%%%%%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
diff changeset
205 end