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annotate +sbp/+implementations/d1_noneq_4.m @ 1198:2924b3a9b921 feature/d2_compatible

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Add OpSet for fully compatible D2Variable, created from regular D2Variable by replacing d1 by first row of D1. Formal reduction by one order of accuracy at the boundary point.
author Martin Almquist <malmquist@stanford.edu>
date Fri, 16 Aug 2019 14:30:28 -0700
parents f7ac3cd6eeaa
children 4cb627c7fb90
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261
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1 function [D1,H,x,h] = d1_noneq_4(N,L)
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2
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3 % L: Domain length
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4 % N: Number of grid points
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5 if(nargin < 2)
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6 L = 1;
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7 end
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8
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9 if(N<8)
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10 error('Operator requires at least 8 grid points');
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11 end
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12
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13 % BP: Number of boundary points
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14 % m: Number of nonequidistant spacings
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15 % order: Accuracy of interior stencil
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16 BP = 4;
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17 m = 2;
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18 order = 4;
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19
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20 %%%% Non-equidistant grid points %%%%%
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21 x0 = 0.0000000000000e+00;
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22 x1 = 6.8764546205559e-01;
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23 x2 = 1.8022115125776e+00;
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24 x3 = 2.8022115125776e+00;
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25 x4 = 3.8022115125776e+00;
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26
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27 xb = sparse(m+1,1);
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28 for i = 0:m
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29 xb(i+1) = eval(['x' num2str(i)]);
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30 end
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31 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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32
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33 %%%% Compute h %%%%%%%%%%
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34 h = L/(2*xb(end) + N-1-2*m);
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35 %%%%%%%%%%%%%%%%%%%%%%%%%
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36
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37 %%%% Define grid %%%%%%%%
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38 x = h*[xb; linspace(xb(end)+1,L/h-xb(end)-1,N-2*(m+1))'; L/h-flip(xb) ];
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39 %%%%%%%%%%%%%%%%%%%%%%%%%
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40
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41 %%%% Norm matrix %%%%%%%%
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42 P = sparse(BP,1);
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43 %#ok<*NASGU>
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44 P0 = 2.1259737557798e-01;
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45 P1 = 1.0260290400758e+00;
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46 P2 = 1.0775123588954e+00;
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47 P3 = 9.8607273802835e-01;
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48
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49 for i = 0:BP-1
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50 P(i+1) = eval(['P' num2str(i)]);
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51 end
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52
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53 H = ones(N,1);
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54 H(1:BP) = P;
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55 H(end-BP+1:end) = flip(P);
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56 H = spdiags(h*H,0,N,N);
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57 %%%%%%%%%%%%%%%%%%%%%%%%%
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58
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59 %%%% Q matrix %%%%%%%%%%%
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60
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61 % interior stencil
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62 switch order
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63 case 2
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64 d = [-1/2,0,1/2];
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65 case 4
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66 d = [1/12,-2/3,0,2/3,-1/12];
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67 case 6
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68 d = [-1/60,3/20,-3/4,0,3/4,-3/20,1/60];
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69 case 8
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70 d = [1/280,-4/105,1/5,-4/5,0,4/5,-1/5,4/105,-1/280];
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71 case 10
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72 d = [-1/1260,5/504,-5/84,5/21,-5/6,0,5/6,-5/21,5/84,-5/504,1/1260];
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73 case 12
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74 d = [1/5544,-1/385,1/56,-5/63,15/56,-6/7,0,6/7,-15/56,5/63,-1/56,1/385,-1/5544];
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75 end
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76 d = repmat(d,N,1);
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77 Q = spdiags(d,-order/2:order/2,N,N);
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78
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79 % Boundaries
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80 Q0_0 = -5.0000000000000e-01;
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81 Q0_1 = 6.5605279837843e-01;
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82 Q0_2 = -1.9875859409017e-01;
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83 Q0_3 = 4.2705795711740e-02;
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84 Q0_4 = 0.0000000000000e+00;
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85 Q0_5 = 0.0000000000000e+00;
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86 Q1_0 = -6.5605279837843e-01;
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87 Q1_1 = 0.0000000000000e+00;
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88 Q1_2 = 8.1236966439895e-01;
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89 Q1_3 = -1.5631686602052e-01;
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90 Q1_4 = 0.0000000000000e+00;
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91 Q1_5 = 0.0000000000000e+00;
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92 Q2_0 = 1.9875859409017e-01;
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93 Q2_1 = -8.1236966439895e-01;
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94 Q2_2 = 0.0000000000000e+00;
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95 Q2_3 = 6.9694440364211e-01;
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96 Q2_4 = -8.3333333333333e-02;
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97 Q2_5 = 0.0000000000000e+00;
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98 Q3_0 = -4.2705795711740e-02;
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99 Q3_1 = 1.5631686602052e-01;
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100 Q3_2 = -6.9694440364211e-01;
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101 Q3_3 = 0.0000000000000e+00;
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102 Q3_4 = 6.6666666666667e-01;
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103 Q3_5 = -8.3333333333333e-02;
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104 for i = 1:BP
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105 for j = 1:BP
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106 Q(i,j) = eval(['Q' num2str(i-1) '_' num2str(j-1)]);
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107 Q(N+1-i,N+1-j) = -eval(['Q' num2str(i-1) '_' num2str(j-1)]);
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108 end
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109 end
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110 %%%%%%%%%%%%%%%%%%%%%%%%%%%
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111
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112 %%%% Difference operator %%
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113 D1 = H\Q;
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114 %%%%%%%%%%%%%%%%%%%%%%%%%%%