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annotate +sbp/higher2_compatible_halfvariable.m @ 85:643bc513b8b8

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Wave2DCurve: Fixed error in tau. Also made random guess at fixing theory from paper.
author Jonatan Werpers <jonatan@werpers.com>
date Mon, 30 Nov 2015 11:41:34 +0100
parents 32b39dc44474
children
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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] = higher2_compatible_halfvariable(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 H=diag(ones(m,1),0);H(1,1)=1/2;H(m,m)=1/2;
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30
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31
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32 H=H*h;
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33 HI=inv(H);
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34
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35
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36 % First derivative SBP operator, 1st order accurate at first 6 boundary points
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37
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38 q1=1/2;
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39 Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
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41 %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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42
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43
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44 e_1=zeros(m,1);e_1(1)=1;
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45 e_m=zeros(m,1);e_m(m)=1;
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46
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47
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48 D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ;
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49
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50 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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51
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52
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53
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54 % Second derivative, 1st order accurate at first boundary points
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55
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56 %% below for constant coefficients
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57 % m1=-1;m0=2;
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58 % 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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59 % M=M/h;
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60 %D2=HI*(-M-e_1*S_1+e_m*S_m);
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61
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62 %% Below for variable coefficients
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63 %% Require a vector c with the koeffients
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64
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65 S_U=[-3/2 2 -1/2]/h;
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66 S_1=zeros(1,m);
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67 S_1(1:3)=S_U;
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68 S_m=zeros(1,m);
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69 S_m(m-2:m)=fliplr(-S_U);
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70
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71 S_1 = S_1';
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72 S_m = S_m';
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73
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74 M=sparse(m,m);
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75 e_1 = sparse(e_1);
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76 e_m = sparse(e_m);
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77 S_1 = sparse(S_1);
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78 S_m = sparse(S_m);
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79
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80 scheme_width = 3;
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81 scheme_radius = (scheme_width-1)/2;
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82 r = (1+scheme_radius):(m-scheme_radius);
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83
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84 function D2 = D2_fun(c)
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85
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86 Mm1 = -c(r-1)/2 - c(r)/2;
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87 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
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88 Mp1 = -c(r)/2 - c(r+1)/2;
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89
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90 M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m);
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91
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92
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93 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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94 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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95 M=M/h;
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96
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97 D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
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98 end
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99 D2 = @D2_fun;
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100
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101
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102
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103
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104
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105 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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106
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107
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108
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109 % Third derivative, 1st order accurate at first 6 boundary points
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110
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111 q2=1/2;q1=-1;
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112 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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113
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114 %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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115
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116
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117 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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118 Q3(1:4,1:4)=Q3_U;
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119 Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) );
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120 Q3=Q3/h^2;
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121
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122
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123
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124 S2_U=[1 -2 1;]/h^2;
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125 S2_1=zeros(1,m);
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126 S2_1(1:3)=S2_U;
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127 S2_m=zeros(1,m);
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128 S2_m(m-2:m)=fliplr(S2_U);
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129 S2_1 = S2_1';
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130 S2_m = S2_m';
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131
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132
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133
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134 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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135
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136 % Fourth derivative, 0th order accurate at first 6 boundary points (still
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137 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
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138
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139 m2=1;m1=-4;m0=6;
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140 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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141
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142 %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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143
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144 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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145
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146
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147 M4(1:4,1:4)=M4_U;
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148
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149 M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) );
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150 M4=M4/h^3;
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151
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152 S3_U=[-1 3 -3 1;]/h^3;
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153 S3_1=zeros(1,m);
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154 S3_1(1:4)=S3_U;
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155 S3_m=zeros(1,m);
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156 S3_m(m-3:m)=fliplr(-S3_U);
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157 S3_1 = S3_1';
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158 S3_m = S3_m';
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159
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160 D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m');
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161 end