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annotate +sbp/+implementations/d2_variable_periodic_4.m @ 1198:2924b3a9b921 feature/d2_compatible

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Add OpSet for fully compatible D2Variable, created from regular D2Variable by replacing d1 by first row of D1. Formal reduction by one order of accuracy at the boundary point.
author Martin Almquist <malmquist@stanford.edu>
date Fri, 16 Aug 2019 14:30:28 -0700
parents bbf303c1f0cf
children
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5ccf6aaf6d6b Add D2VariablePeriodic orders 4 and 6.
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_periodic_4(m,h)
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Martin Almquist <malmquist@stanford.edu>
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2 % m = number of unique grid points, i.e. h = L/m;
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3
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Martin Almquist <malmquist@stanford.edu>
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4 if(m<5)
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5 error(['Operator requires at least ' num2str(5) ' grid points']);
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6 end
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7
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8 % Norm
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Martin Almquist <malmquist@stanford.edu>
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9 Hv = ones(m,1);
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Martin Almquist <malmquist@stanford.edu>
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10 Hv = h*Hv;
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Martin Almquist <malmquist@stanford.edu>
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11 H = spdiag(Hv, 0);
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Martin Almquist <malmquist@stanford.edu>
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12 HI = spdiag(1./Hv, 0);
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Martin Almquist <malmquist@stanford.edu>
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15 % Dummy boundary operators
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Martin Almquist <malmquist@stanford.edu>
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16 e_l = sparse(m,1);
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Martin Almquist <malmquist@stanford.edu>
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17 e_r = rot90(e_l, 2);
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18
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Martin Almquist <malmquist@stanford.edu>
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19 d1_l = sparse(m,1);
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20 d1_r = -rot90(d1_l, 2);
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21
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Martin Almquist <malmquist@stanford.edu>
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22 S = d1_l*d1_l' + d1_r*d1_r';
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23
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Martin Almquist <malmquist@stanford.edu>
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24 % D1 operator
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Martin Almquist <malmquist@stanford.edu>
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25 stencil = [1/12 -2/3 0 2/3 -1/12];
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Martin Almquist <malmquist@stanford.edu>
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26 diags = -2:2;
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Martin Almquist <malmquist@stanford.edu>
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27 Q = stripeMatrixPeriodic(stencil, diags, m);
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Martin Almquist <malmquist@stanford.edu>
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28 D1 = HI*(Q - 1/2*e_l*e_l' + 1/2*e_r*e_r');
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29
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30
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Martin Almquist <malmquist@stanford.edu>
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31 scheme_width = 5;
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Martin Almquist <malmquist@stanford.edu>
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32 scheme_radius = (scheme_width-1)/2;
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Jonatan Werpers <jonatan@werpers.com>
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33
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34 r = 1:m;
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Martin Almquist <malmquist@stanford.edu>
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35 offset = scheme_width;
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36 r = r + offset;
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37
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38 function D2 = D2_fun(c)
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39 c = [c(end-scheme_width+1:end); c; c(1:scheme_width) ];
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40
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Martin Almquist <malmquist@stanford.edu>
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41 % Note: these coefficients are for -M.
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Martin Almquist <malmquist@stanford.edu>
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42 Mm2 = -1/8*c(r-2) + 1/6*c(r-1) - 1/8*c(r);
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Martin Almquist <malmquist@stanford.edu>
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43 Mm1 = 1/6 *c(r-2) + 1/2*c(r-1) + 1/2*c(r) + 1/6*c(r+1);
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Martin Almquist <malmquist@stanford.edu>
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44 M0 = -1/24*c(r-2)- 5/6*c(r-1) - 3/4*c(r) - 5/6*c(r+1) - 1/24*c(r+2);
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Martin Almquist <malmquist@stanford.edu>
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45 Mp1 = 0 * c(r-2) + 1/6*c(r-1) + 1/2*c(r) + 1/2*c(r+1) + 1/6 *c(r+2);
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46 Mp2 = 0 * c(r-2) + 0 * c(r-1) - 1/8*c(r) + 1/6*c(r+1) - 1/8 *c(r+2);
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47
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Martin Almquist <malmquist@stanford.edu>
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48 vals = -[Mm2,Mm1,M0,Mp1,Mp2];
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49 diags = -scheme_radius : scheme_radius;
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50 M = spdiagsPeriodic(vals,diags);
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51
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52 M=M/h;
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53 D2=HI*(-M );
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54
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55 end
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Martin Almquist <malmquist@stanford.edu>
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56 D2 = @D2_fun;
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57 end