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Fixed formatting.
author Jonatan Werpers <jonatan@werpers.com>
date Fri, 23 Sep 2016 19:14:04 +0200
parents 713b125038a3
children 52b4cdf27633
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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1 % Returns D2 as a function handle
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2 function [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = 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 BP = 4;
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13 if(m<2*BP)
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14 error(['Operator requires at least ' num2str(2*BP) ' grid points']);
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15 end
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16
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17
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18 H=speye(m,m);
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19 H(1,1)=1/2;
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20 H(m,m)=1/2;
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21
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22 H=H*h;
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23 HI=inv(H);
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24
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25
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26 % First derivative SBP operator, 1st order accurate at first 6 boundary points
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27
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28 q1=1/2;
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29 % Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
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30 stencil = [-q1,0,q1];
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31 d = (length(stencil)-1)/2;
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32 diags = -d:d;
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33 Q = stripeMatrix(stencil, diags, m);
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34
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35 %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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36
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37 e_1=sparse(m,1);
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38 e_1(1)=1;
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39 e_m=sparse(m,1);
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40 e_m(m)=1;
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41
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42 D1=HI*(Q-1/2*(e_1*e_1')+1/2*(e_m*e_m')) ;
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43
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44
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45 % Second derivative, 1st order accurate at first boundary points
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46
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47 % below for constant coefficients
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48 % m1=-1;m0=2;
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49 % 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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50 % M=M/h;
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51 %D2=HI*(-M-e_1*S_1+e_m*S_m);
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52
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53 % Below for variable coefficients
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54 % Require a vector c with the koeffients
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55
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56 S_U=[-3/2 2 -1/2]/h;
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57 S_1=sparse(1,m);
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58 S_1(1:3)=S_U;
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59 S_m=sparse(1,m);
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60 S_m(m-2:m)=fliplr(-S_U);
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61
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62 S_1 = S_1';
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63 S_m = S_m';
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64
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65 M=sparse(m,m);
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66 e_1 = sparse(e_1);
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67 e_m = sparse(e_m);
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68 S_1 = sparse(S_1);
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69 S_m = sparse(S_m);
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70
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71 scheme_width = 3;
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72 scheme_radius = (scheme_width-1)/2;
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73 r = (1+scheme_radius):(m-scheme_radius);
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74
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75 function D2 = D2_fun(c)
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76
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77 Mm1 = -c(r-1)/2 - c(r)/2;
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78 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
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79 Mp1 = -c(r)/2 - c(r+1)/2;
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80
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81 M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m);
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82
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83
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84 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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85 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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86 M=M/h;
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87
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88 D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
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89 end
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90 D2 = @D2_fun;
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91
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92
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93
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94
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95
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96 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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97
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98
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99
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100 % Third derivative, 1st order accurate at first 6 boundary points
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101
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102 q2=1/2;q1=-1;
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103 % 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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104 stencil = [-q2,-q1,0,q1,q2];
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105 d = (length(stencil)-1)/2;
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106 diags = -d:d;
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107 Q3 = stripeMatrix(stencil, diags, m);
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108
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109 %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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110
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111
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112 Q3_U = [
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113 0 -0.13e2/0.16e2 0.7e1/0.8e1 -0.1e1/0.16e2;
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114 0.13e2/0.16e2 0 -0.23e2/0.16e2 0.5e1/0.8e1;
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115 -0.7e1/0.8e1 0.23e2/0.16e2 0 -0.17e2/0.16e2;
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116 0.1e1/0.16e2 -0.5e1/0.8e1 0.17e2/0.16e2 0;
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117 ];
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118 Q3(1:4,1:4)=Q3_U;
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119 Q3(m-3:m,m-3:m)=rot90( -Q3_U ,2 );
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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=sparse(1,m);
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126 S2_1(1:3)=S2_U;
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127 S2_m=sparse(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') ) ;
261
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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 stencil = [m2,m1,m0,m1,m2];
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142 d = (length(stencil)-1)/2;
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143 diags = -d:d;
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144 M4 = stripeMatrix(stencil, diags, m);
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145
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146 %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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147
312
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148 M4_U=[
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149 0.13e2/0.10e2 -0.12e2/0.5e1 0.9e1/0.10e2 0.1e1/0.5e1;
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150 -0.12e2/0.5e1 0.26e2/0.5e1 -0.16e2/0.5e1 0.2e1/0.5e1;
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151 0.9e1/0.10e2 -0.16e2/0.5e1 0.47e2/0.10e2 -0.17e2/0.5e1;
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152 0.1e1/0.5e1 0.2e1/0.5e1 -0.17e2/0.5e1 0.29e2/0.5e1;
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153 ];
261
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154
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155 M4(1:4,1:4)=M4_U;
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156
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157 M4(m-3:m,m-3:m)=rot90( M4_U ,2 );
261
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158 M4=M4/h^3;
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159
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160 S3_U=[-1 3 -3 1;]/h^3;
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161 S3_1=sparse(1,m);
261
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162 S3_1(1:4)=S3_U;
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163 S3_m=sparse(1,m);
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164 S3_m(m-3:m)=fliplr(-S3_U);
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165 S3_1 = S3_1';
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166 S3_m = S3_m';
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167
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168 D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m');
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169 end