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Make the hollow scheme generation more efficient by introducing the D2VariableHollow opSet
author Martin Almquist <malmquist@stanford.edu>
date Sat, 11 Jul 2020 06:54:15 -0700
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children b5907140c069
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49e3870335ef Make the hollow scheme generation more efficient by introducing the D2VariableHollow opSet
Martin Almquist <malmquist@stanford.edu>
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1 function [H, HI, D1, D2, e_l, e_r, d1_l, d1_r] = d2_variable_hollow_2(m,h)
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2
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3 BP = 1;
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4 if(m<2*BP)
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5 error(['Operator requires at least ' num2str(2*BP) ' grid points']);
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6 end
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7
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8 % Norm
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9 Hv = ones(m,1);
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10 Hv(1) = 1/2;
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11 Hv(m:m) = 1/2;
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12 Hv = h*Hv;
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13 H = spdiag(Hv, 0);
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14 HI = spdiag(1./Hv, 0);
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16
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17 % Boundary operators
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18 e_l = sparse(m,1);
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19 e_l(1) = 1;
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20 e_r = rot90(e_l, 2);
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21
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22 d1_l = sparse(m,1);
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23 d1_l(1:3) = 1/h*[-3/2 2 -1/2];
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24 d1_r = -rot90(d1_l, 2);
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25
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26 % D1 operator
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27 diags = -1:1;
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28 stencil = [-1/2 0 1/2];
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29 D1 = stripeMatrix(stencil, diags, m);
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30
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31 D1(1,1)=-1;D1(1,2)=1;D1(m,m-1)=-1;D1(m,m)=1;
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32 D1(m,m-1)=-1;D1(m,m)=1;
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33 D1=D1/h;
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34 %Q=H*D1 + 1/2*(e_1*e_1') - 1/2*(e_m*e_m');
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35
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36
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37 M=sparse(m,m);
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38 nBP = 2;
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39
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40 scheme_width = 3;
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41 scheme_radius = (scheme_width-1)/2;
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42
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43 function D2 = D2_fun(c)
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44
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45 for r = 1+scheme_radius:nBP
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46 Mm1 = -c(r-1)/2 - c(r)/2;
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47 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
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48 Mp1 = -c(r)/2 - c(r+1)/2;
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49
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50 M(r, r-1) = Mm1;
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51 M(r, r) = M0;
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52 M(r, r+1) = Mp1;
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53 end
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54
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55 for r = m-nBP+1:m-scheme_radius
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56 Mm1 = -c(r-1)/2 - c(r)/2;
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57 M0 = c(r-1)/2 + c(r) + c(r+1)/2;
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58 Mp1 = -c(r)/2 - c(r+1)/2;
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59
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60 M(r, r-1) = Mm1;
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61 M(r, r) = M0;
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62 M(r, r+1) = Mp1;
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63 end
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64
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65 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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66 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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67 M=M/h;
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68
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69 D2=HI*(-M-c(1)*e_l*d1_l'+c(m)*e_r*d1_r');
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70 end
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71 D2 = @D2_fun;
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72 end