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Fixed function names in renamed files.
author Jonatan Werpers <jonatan@werpers.com>
date Fri, 23 Sep 2016 14:59:55 +0200
parents ffa5d557942b
children 9230c056a574
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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Martin Almquist <martin.almquist@it.uu.se>
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1 % Returns D2 as a function handle
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2 function [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = d4_variable_2(m,h)
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3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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4 %%% 4:de ordn. SBP Finita differens %%%
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5 %%% operatorer framtagna av Ken Mattsson %%%
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6 %%% %%%
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7 %%% 6 randpunkter, diagonal norm %%%
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8 %%% %%%
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9 %%% Datum: 2013-11-11 %%%
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10 %%% %%%
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11 %%% %%%
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12 %%% H (Normen) %%%
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13 %%% D1 (approx f?rsta derivatan) %%%
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14 %%% D2 (approx andra derivatan) %%%
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15 %%% D3 (approx tredje derivatan) %%%
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16 %%% D2 (approx fj?rde derivatan) %%%
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17 %%% %%%
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18 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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19
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20 % M?ste ange antal punkter (m) och stegl?ngd (h)
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21 % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
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22 % vi har 3de och 4de derivator i v?r PDE
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23 % I annat fall anv?nd de "traditionella" som har noggrannare
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24 % randsplutningar f?r D1 och D2
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25
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26 % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
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27 % vilket ?r n?dv?ndigt f?r stabilitet
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28
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29 BP = 4;
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30 if(m<2*BP)
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31 error(['Operator requires at least ' num2str(2*BP) ' grid points']);
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32 end
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33
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34 H=speye(m,m);H(1,1)=1/2;H(m,m)=1/2;
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35
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36
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37 H=H*h;
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38 HI=inv(H);
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39
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40
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41 % First derivative SBP operator, 1st order accurate at first 6 boundary points
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42
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43 q1=1/2;
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44 % Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
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45 stencil = [-q1,0,q1];
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46 d = (length(stencil)-1)/2;
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47 diags = -d:d;
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48 Q = stripeMatrix(stencil, diags, m);
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49
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50 %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2));
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51
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52
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53 e_1=sparse(m,1);e_1(1)=1;
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54 e_m=sparse(m,1);e_m(m)=1;
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55
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56
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57 D1=HI*(Q-1/2*(e_1*e_1')+1/2*(e_m*e_m')) ;
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58
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59 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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60
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61
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62
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63 % Second derivative, 1st order accurate at first boundary points
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64
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65 % below for constant coefficients
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66 % m1=-1;m0=2;
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67 % M=m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);M(1,1)=1;M(m,m)=1;
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68 % M=M/h;
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69 %D2=HI*(-M-e_1*S_1+e_m*S_m);
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70
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71 % Below for variable coefficients
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72 % Require a vector c with the koeffients
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73
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74 S_U=[-3/2 2 -1/2]/h;
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75 S_1=sparse(1,m);
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76 S_1(1:3)=S_U;
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77 S_m=sparse(1,m);
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78 S_m(m-2:m)=fliplr(-S_U);
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79
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80 S_1 = S_1';
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81 S_m = S_m';
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82
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83 M=sparse(m,m);
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84 e_1 = sparse(e_1);
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85 e_m = sparse(e_m);
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86 S_1 = sparse(S_1);
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87 S_m = sparse(S_m);
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88
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89 scheme_width = 3;
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90 scheme_radius = (scheme_width-1)/2;
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91 r = (1+scheme_radius):(m-scheme_radius);
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92
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93 function D2 = D2_fun(c)
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94
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95 Mm1 = -c(r-1)/2 - c(r)/2;
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96 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
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97 Mp1 = -c(r)/2 - c(r+1)/2;
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98
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99 M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m);
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100
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101
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102 M(1:2,1:2)=[c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;];
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103 M(m-1:m,m-1:m)=[c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;];
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104 M=M/h;
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105
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106 D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
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107 end
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108 D2 = @D2_fun;
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109
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110
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111
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112
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113
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114 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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115
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116
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117
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118 % Third derivative, 1st order accurate at first 6 boundary points
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119
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120 q2=1/2;q1=-1;
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121 % Q3=q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
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122 stencil = [-q2,-q1,0,q1,q2];
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123 d = (length(stencil)-1)/2;
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124 diags = -d:d;
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125 Q3 = stripeMatrix(stencil, diags, m);
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126
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127 %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3));
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128
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129
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130 Q3_U = [0 -0.13e2 / 0.16e2 0.7e1 / 0.8e1 -0.1e1 / 0.16e2; 0.13e2 / 0.16e2 0 -0.23e2 / 0.16e2 0.5e1 / 0.8e1; -0.7e1 / 0.8e1 0.23e2 / 0.16e2 0 -0.17e2 / 0.16e2; 0.1e1 / 0.16e2 -0.5e1 / 0.8e1 0.17e2 / 0.16e2 0;];
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131 Q3(1:4,1:4)=Q3_U;
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132 Q3(m-3:m,m-3:m)=rot90( -Q3_U ,2 );
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133 Q3=Q3/h^2;
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134
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135
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136
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137 S2_U=[1 -2 1;]/h^2;
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138 S2_1=sparse(1,m);
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139 S2_1(1:3)=S2_U;
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140 S2_m=sparse(1,m);
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141 S2_m(m-2:m)=fliplr(S2_U);
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142 S2_1 = S2_1';
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143 S2_m = S2_m';
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144
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145
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146
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147 D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*(S_1*S_1') -1/2*(S_m*S_m') ) ;
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148
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149 % Fourth derivative, 0th order accurate at first 6 boundary points (still
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150 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
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151
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152 m2=1;m1=-4;m0=6;
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153 % M4=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);
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154 stencil = [m2,m1,m0,m1,m2];
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155 d = (length(stencil)-1)/2;
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156 diags = -d:d;
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157 M4 = stripeMatrix(stencil, diags, m);
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158
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159 %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0));
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160
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161 M4_U=[0.13e2 / 0.10e2 -0.12e2 / 0.5e1 0.9e1 / 0.10e2 0.1e1 / 0.5e1; -0.12e2 / 0.5e1 0.26e2 / 0.5e1 -0.16e2 / 0.5e1 0.2e1 / 0.5e1; 0.9e1 / 0.10e2 -0.16e2 / 0.5e1 0.47e2 / 0.10e2 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;];
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162
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163
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164 M4(1:4,1:4)=M4_U;
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165
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166 M4(m-3:m,m-3:m)=rot90( M4_U ,2 );
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167 M4=M4/h^3;
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168
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169 S3_U=[-1 3 -3 1;]/h^3;
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170 S3_1=sparse(1,m);
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171 S3_1(1:4)=S3_U;
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172 S3_m=sparse(1,m);
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173 S3_m(m-3:m)=fliplr(-S3_U);
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174 S3_1 = S3_1';
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175 S3_m = S3_m';
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176
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177 D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m');
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178 end