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annotate +sbp/higher_compatible2.m @ 87:0a29a60e0b21

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In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
author Martin Almquist <martin.almquist@it.uu.se>
date Sun, 29 Nov 2015 22:23:09 +0100
parents 32b39dc44474
children
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1 function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_compatible2(m,h)
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2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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3 %%% 4:de ordn. SBP Finita differens %%%
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4 %%% operatorer framtagna av Ken Mattsson %%%
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5 %%% %%%
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6 %%% 6 randpunkter, diagonal norm %%%
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7 %%% %%%
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8 %%% Datum: 2013-11-11 %%%
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9 %%% %%%
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10 %%% %%%
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11 %%% H (Normen) %%%
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12 %%% D1 (approx f?rsta derivatan) %%%
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13 %%% D2 (approx andra derivatan) %%%
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14 %%% D3 (approx tredje derivatan) %%%
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15 %%% D2 (approx fj?rde derivatan) %%%
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16 %%% %%%
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17 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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18
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19 % M?ste ange antal punkter (m) och stegl?ngd (h)
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20 % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
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21 % vi har 3de och 4de derivator i v?r PDE
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22 % I annat fall anv?nd de "traditionella" som har noggrannare
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23 % randsplutningar f?r D1 och D2
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24
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25 % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
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26 % vilket ?r n?dv?ndigt f?r stabilitet
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27
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28 H=diag(ones(m,1),0);H(1,1)=1/2;H(m,m)=1/2;
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29
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30
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31 H=H*h;
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32 HI=inv(H);
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33
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34
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35 % First derivative SBP operator, 1st order accurate at first 6 boundary points
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36
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37 q1=1/2;
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38 Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
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39
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40 %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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41
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42
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43 e_1=zeros(m,1);e_1(1)=1;
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44 e_m=zeros(m,1);e_m(m)=1;
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45
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46
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47 D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ;
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48
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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 % Second derivative, 1st order accurate at first 6 boundary points
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54 m1=-1;m0=2;
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55 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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56 M=M/h;
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57
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58 S_U=[-1 1]/h;
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59 S_1=zeros(1,m);
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60 S_1(1:2)=S_U;
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61 S_m=zeros(1,m);
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62
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63 S_m(m-1:m)=fliplr(-S_U);
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64
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65 D2=HI*(-M-e_1*S_1+e_m*S_m);
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66
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67
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68 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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69
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70
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71
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72 % Third derivative, 1st order accurate at first 6 boundary points
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73
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74 q2=1/2;q1=-1;
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75 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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76
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77 %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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78
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79
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80 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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81 Q3(1:4,1:4)=Q3_U;
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82 Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) );
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83 Q3=Q3/h^2;
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84
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85
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86
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87 S2_U=[1 -2 1;]/h^2;
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88 S2_1=zeros(1,m);
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89 S2_1(1:3)=S2_U;
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90 S2_m=zeros(1,m);
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91 S2_m(m-2:m)=fliplr(S2_U);
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92
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94
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95 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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96
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97 % Fourth derivative, 0th order accurate at first 6 boundary points (still
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98 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
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99
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100 m2=1;m1=-4;m0=6;
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101 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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102
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103 %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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104
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105 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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106
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107 M4(1:4,1:4)=M4_U;
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108
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109 M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) );
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110 M4=M4/h^3;
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111
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112 S3_U=[-1 3 -3 1;]/h^3;
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113 S3_1=zeros(1,m);
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114 S3_1(1:4)=S3_U;
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115 S3_m=zeros(1,m);
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116 S3_m(m-3:m)=fliplr(-S3_U);
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117
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118 D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m);
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119
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32b39dc44474 Removed repository inside +sbp to make it part of the root repo.
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121
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122 S_1 = S_1';
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123 S_m = S_m';
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124 S2_1 = S2_1';
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125 S2_m = S2_m';
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126 S3_1 = S3_1';
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127 S3_m = S3_m';
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128
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129
32b39dc44474 Removed repository inside +sbp to make it part of the root repo.
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130
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132 % L=h*(m-1);
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133
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134 % x1=linspace(0,L,m)';
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135 % x2=x1.^2/fac(2);
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136 % x3=x1.^3/fac(3);
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137 % x4=x1.^4/fac(4);
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138 % x5=x1.^5/fac(5);
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139
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140 % x0=x1.^0/fac(1);
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141
32b39dc44474 Removed repository inside +sbp to make it part of the root repo.
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142
32b39dc44474 Removed repository inside +sbp to make it part of the root repo.
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143 end