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annotate +sbp/+implementations/d2_noneq_variable_4.m @ 1344:b4e5e45bd239 feature/D2_boundary_opt

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Remove round off zeros from D2Nonequidistant operators
author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Sat, 15 Oct 2022 15:48:20 +0200
parents 8e9df030a0a5
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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_4(N, h, options)
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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2 % N: Number of grid points
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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3 % h: grid spacing
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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4 % options: struct containing options for constructing the operator
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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5 % current options are:
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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6 % options.stencil_type ('minimal','nonminimal','wide')
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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:
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8
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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9 % BP: Number of boundary points
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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10 % order: Accuracy of interior stencil
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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11 BP = 4;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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12 order = 4;
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855871e0b852 Add size checks to the operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1326
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13 if(N<2*BP)
855871e0b852 Add size checks to the operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1326
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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: 1326
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15 end
1325
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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16
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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17 %%%% Norm matrix %%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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18 P = zeros(BP, 1);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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19 P0 = 2.1259737557798e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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20 P1 = 1.0260290400758e+00;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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21 P2 = 1.0775123588954e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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22 P3 = 9.8607273802835e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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23
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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24 for i = 0:BP - 1
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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25 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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26 end
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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27
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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28 Hv = ones(N, 1);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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29 Hv(1:BP) = P;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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30 Hv(end - BP + 1:end) = flip(P);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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31 Hv = h * Hv;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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32 H = spdiags(Hv, 0, N, N);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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33 HI = spdiags(1 ./ Hv, 0, N, N);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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34 %%%%%%%%%%%%%%%%%%%%%%%%%
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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35
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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36 %%%% Q matrix %%%%%%%%%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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37 d = [1/12, -2/3, 0, 2/3, -1/12];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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38 d = repmat(d, N, 1);
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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39 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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40
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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41 % Boundaries
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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42 Q0_0 = -5.0000000000000e-01;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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43 Q0_1 = 6.5605279837843e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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44 Q0_2 = -1.9875859409017e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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45 Q0_3 = 4.2705795711740e-02;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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46 Q0_4 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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47 Q0_5 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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48 Q1_0 = -6.5605279837843e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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49 Q1_1 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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50 Q1_2 = 8.1236966439895e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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51 Q1_3 = -1.5631686602052e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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52 Q1_4 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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53 Q1_5 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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54 Q2_0 = 1.9875859409017e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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55 Q2_1 = -8.1236966439895e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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56 Q2_2 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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57 Q2_3 = 6.9694440364211e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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58 Q2_4 = -8.3333333333333e-02;
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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59 Q2_5 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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60 Q3_0 = -4.2705795711740e-02;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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61 Q3_1 = 1.5631686602052e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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62 Q3_2 = -6.9694440364211e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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63 Q3_3 = 0.0000000000000e+00;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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64 Q3_4 = 6.6666666666667e-01;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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65 Q3_5 = -8.3333333333333e-02;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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66
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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67 for i = 1:BP
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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68
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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69 for j = 1:BP
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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70 Q(i, 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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71 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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72 end
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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73
1b0f2415237f Add variable coefficient boundary-optimized second derivatives.
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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74 end
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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75
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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76 %%%%%%%%%%%%%%%%%%%%%%%%%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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77
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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78 %%% Undivided difference operators %%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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79 % Closed with zeros at the first boundary nodes.
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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80 m = N;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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81
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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82 DD_2 = (diag(ones(m - 1, 1), -1) - 2 * diag(ones(m, 1), 0) + diag(ones(m - 1, 1), 1));
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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83 DD_2(1:3, 1:4) = [0 0 0 0; 0.16138369498429727170e1 -0.26095138364100825853e1 0.99567688656710986834e0 0; 0 0.84859980956172494512e0 -0.17944203477786665350e1 0.94582053821694158989e0; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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84 DD_2(m - 2:m, m - 3:m) = [0.94582053821694158989e0 -0.17944203477786665350e1 0.84859980956172494512e0 0; 0 0.99567688656710986834e0 -0.26095138364100825853e1 0.16138369498429727170e1; 0 0 0 0; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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85 DD_2 = sparse(DD_2);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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86
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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87 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));
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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88 DD_3(1:4, 1:5) = [0 0 0 0 0; 0 0 0 0 0; -0.17277463987989539852e1 0.37021976718569105700e1 -0.29870306597013296050e1 0.10125793866433730203e1 0; 0 -0.81738495424057284493e0 0.26916305216679998025e1 -0.28374616146508247697e1 0.96321604722339781208e0; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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89 DD_3(m - 2:m, m - 4:m) = [-0.96321604722339781208e0 0.28374616146508247697e1 -0.26916305216679998025e1 0.81738495424057284493e0 0; 0 -0.10125793866433730203e1 0.29870306597013296050e1 -0.37021976718569105700e1 0.17277463987989539852e1; 0 0 0 0 0; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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90 DD_3 = sparse(DD_3);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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91
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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92 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));
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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93 DD_4(1:4, 1:6) = [0 0 0 0 0 0; 0 0 0 0 0 0; 0.18176226052481525189e1 -0.47546882767009058782e1 0.59740613194026592100e1 -0.40503175465734920811e1 0.10133218986235862303e1 0; 0 0.79462567299107735362e0 -0.35888406955573330700e1 0.56749232293016495393e1 -0.38528641888935912483e1 0.97215598215819742539e0; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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94 DD_4(m - 3:m, m - 5:m) = [0.97215598215819742539e0 -0.38528641888935912483e1 0.56749232293016495393e1 -0.35888406955573330700e1 0.79462567299107735362e0 0; 0 0.10133218986235862303e1 -0.40503175465734920811e1 0.59740613194026592100e1 -0.47546882767009058782e1 0.18176226052481525189e1; 0 0 0 0 0 0; 0 0 0 0 0 0; ];
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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95 DD_4 = sparse(DD_4);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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96 %%%%%%%%%%%%%%%%%%%%%%%%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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97
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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98 %%%% Difference operators %%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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99 D1 = H \ Q;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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100
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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101 % Helper functions for constructing D2(c)
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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102 % TODO: Consider changing sparse(diag(...)) to spdiags(....)
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b4e5e45bd239 Remove round off zeros from D2Nonequidistant operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1332
diff changeset
103 min_inds = sbp.implementations.d2_sparsity_pattern_inds(m, order, BP, 0, 2);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1332
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104 nonmin_inds = sbp.implementations.d2_sparsity_pattern_inds(m, order, BP, 1, 2);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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105
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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106 % Minimal 5 point stencil width
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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107 function D2 = D2_fun_minimal(c)
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108 % Here we add variable diffusion
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109 C1 = sparse(diag(c));
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110 C2 = 1/2 * diag(ones(m - 1, 1), -1) + 1/2 * diag(ones(m, 1), 0); C2(1, 2) = 1/2;
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111
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112 C2 = sparse(diag(C2 * c));
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113
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114 % Remainder term added to wide second drivative opereator, to obtain a 5
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115 % point narrow stencil.
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116 R = (1/144 / h) * transpose(DD_4) * C1 * DD_4 + (1/18 / h) * transpose(DD_3) * C2 * DD_3;
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117 D2 = D1 * C1 * D1 - H \ R;
1344
b4e5e45bd239 Remove round off zeros from D2Nonequidistant operators
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1332
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118
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1332
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119 % Remove potential round off zeros
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1332
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120 D2tmp = sparse(m,m);
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parents: 1332
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121 D2tmp(min_inds) = D2(min_inds);
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parents: 1332
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122 D2 = D2tmp;
1325
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123 end
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124
1332
8e9df030a0a5 Clarify comments
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1330
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125 % Non-minimal 7 point stencil width
1325
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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126 function D2 = D2_fun_nonminimal(c)
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127 % Here we add variable diffusion
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128 C1 = sparse(diag(c));
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129
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130 % Remainder term added to wide second derivative operator
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131 R = (1/144 / h) * transpose(DD_4) * C1 * DD_4;
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132 D2 = D1 * C1 * D1 - H \ R;
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parents: 1332
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133
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parents: 1332
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134 % Remove potential round off zeros
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parents: 1332
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135 D2tmp = sparse(m,m);
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136 D2tmp(nonmin_inds) = D2(nonmin_inds);
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parents: 1332
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137 D2 = D2tmp;
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138 end
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139
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140 % Wide stencil
1326
c2d716c4f1ed Fix bug when using wide stencils
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents: 1325
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141 function D2 = D2_fun_wide(c)
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142 % Here we add variable diffusion
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143 C1 = sparse(diag(c));
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144 D2 = D1 * C1 * D1;
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145 end
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146
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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147 switch options.stencil_width
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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148 case 'minimal'
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149 D2 = @D2_fun_minimal;
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150 case 'nonminimal'
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151 D2 = @D2_fun_nonminimal;
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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152 case 'wide'
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153 D2 = @D2_fun_wide;
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154 otherwise
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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155 error('No option %s for stencil width', options.stencil_width)
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156 end
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parents:
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157
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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158 %%%%%%%%%%%%%%%%%%%%%%%%%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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159
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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160 %%%% Artificial dissipation operator %%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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161 switch options.AD
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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162 case 'upwind'
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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163 % This is the choice that yield 3rd order Upwind
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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164 DI = H \ (transpose(DD_2) * DD_2) * (-1/12);
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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165 case 'op'
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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166 % This choice will preserve the order of the underlying
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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167 % Non-dissipative D1 SBP operator
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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168 DI = H \ (transpose(DD_3) * DD_3) * (-1 / (5 * 12));
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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169 otherwise
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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170 error("Artificial dissipation options '%s' not implemented.", option.AD)
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parents:
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171 end
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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172
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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173 %%%%%%%%%%%%%%%%%%%%%%%%%%%
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
parents:
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174 end