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annotate +sbp/+implementations/d4_compatible_2.m @ 1337:bf2554f1825d feature/D2_boundary_opt

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Add periodic D1 and D2 operators for orders 8,10,12
author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Fri, 13 May 2022 13:28:10 +0200
parents f7ac3cd6eeaa
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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1 function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m,...
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2 S3_1, S3_m, S_1, S_m] = d4_compatible_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=H\(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 6 boundary points
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64 % m1=-1;m0=2;
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65 % % 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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66 % stencil = [m2,m1,m0,m1,m2];
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67 % d = (length(stencil)-1)/2;
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68 % diags = -d:d;
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69 % M = stripeMatrix(stencil, diags, m);
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70 % M=M/h;
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71
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72 S_U=[-1 1]/h;
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73 S_1=sparse(1,m);
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74 S_1(1:2)=S_U;
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75 S_m=sparse(1,m);
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76
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77 S_m(m-1:m)=fliplr(-S_U);
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78
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79 % D2=H\(-M-e_1*S_1+e_m*S_m);
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80
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81
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82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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83
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84
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85
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86 % Third derivative, 1st order accurate at first 6 boundary points
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87
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88 % q2=1/2;q1=-1;
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89 % % 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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90 % stencil = [-q2,-q1,0,q1,q2];
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91 % d = (length(stencil)-1)/2;
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92 % diags = -d:d;
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93 % Q3 = stripeMatrix(stencil, diags, m);
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94
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95 %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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96
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97
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98 % Q3_U = [0 -0.2e1 / 0.5e1 0.3e1 / 0.10e2 0.1e1 / 0.10e2; 0.2e1 / 0.5e1 0 -0.7e1 / 0.10e2 0.3e1 / 0.10e2; -0.3e1 / 0.10e2 0.7e1 / 0.10e2 0 -0.9e1 / 0.10e2; -0.1e1 / 0.10e2 -0.3e1 / 0.10e2 0.9e1 / 0.10e2 0;];
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99 % Q3(1:4,1:4)=Q3_U;
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100 % Q3(m-3:m,m-3:m)=rot90( -Q3_U ,2 );
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101 % Q3=Q3/h^2;
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102
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103
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104
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105 S2_U=[1 -2 1;]/h^2;
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106 S2_1=sparse(1,m);
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107 S2_1(1:3)=S2_U;
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108 S2_m=sparse(1,m);
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109 S2_m(m-2:m)=fliplr(S2_U);
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110
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112
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113 % D3=H\(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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114
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115 % Fourth derivative, 0th order accurate at first 6 boundary points (still
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116 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
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117
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118 m2=1;m1=-4;m0=6;
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119 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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120
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121 %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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122
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123 M4_U=[0.4e1 / 0.5e1 -0.7e1 / 0.5e1 0.2e1 / 0.5e1 0.1e1 / 0.5e1; -0.7e1 / 0.5e1 0.16e2 / 0.5e1 -0.11e2 / 0.5e1 0.2e1 / 0.5e1; 0.2e1 / 0.5e1 -0.11e2 / 0.5e1 0.21e2 / 0.5e1 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;];
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124
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125 M4(1:4,1:4)=M4_U;
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126
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127 M4(m-3:m,m-3:m)=rot90( M4_U ,2 );
261
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128 M4=M4/h^3;
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129
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130 S3_U=[-1 3 -3 1;]/h^3;
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131 S3_1=sparse(1,m);
261
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132 S3_1(1:4)=S3_U;
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133 S3_m=sparse(1,m);
261
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134 S3_m(m-3:m)=fliplr(-S3_U);
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135
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136 D4=H\(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m);
261
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137
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138
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139
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140 S_1 = S_1';
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141 S_m = S_m';
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142 S2_1 = S2_1';
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143 S2_m = S2_m';
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144 S3_1 = S3_1';
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145 S3_m = S3_m';
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146
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147
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148
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149
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150 % L=h*(m-1);
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151
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152 % x1=linspace(0,L,m)';
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153 % x2=x1.^2/fac(2);
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154 % x3=x1.^3/fac(3);
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155 % x4=x1.^4/fac(4);
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156 % x5=x1.^5/fac(5);
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157
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158 % x0=x1.^0/fac(1);
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159
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160
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161 end