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annotate +sbp/D2VariableCompatible.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
children a3d9567d9004
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1198
2924b3a9b921 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.
Martin Almquist <malmquist@stanford.edu>
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1 classdef D2VariableCompatible < sbp.OpSet
2924b3a9b921 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.
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2 properties
2924b3a9b921 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.
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3 D1 % SBP operator approximating first derivative
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4 H % Norm matrix
2924b3a9b921 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.
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5 HI % H^-1
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6 Q % Skew-symmetric matrix
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7 e_l % Left boundary operator
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8 e_r % Right boundary operator
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9 D2 % SBP operator for second derivative
2924b3a9b921 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.
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10 M % Norm matrix, second derivative
2924b3a9b921 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.
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11 d1_l % Left boundary first derivative
2924b3a9b921 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.
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12 d1_r % Right boundary first derivative
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13 m % Number of grid points.
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14 h % Step size
2924b3a9b921 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.
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15 x % grid
2924b3a9b921 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.
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16 borrowing % Struct with borrowing limits for different norm matrices
2924b3a9b921 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.
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17 end
2924b3a9b921 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.
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18
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19 methods
2924b3a9b921 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.
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20 function obj = D2VariableCompatible(m,lim,order)
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21
2924b3a9b921 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.
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22 x_l = lim{1};
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23 x_r = lim{2};
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24 L = x_r-x_l;
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25 obj.h = L/(m-1);
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26 obj.x = linspace(x_l,x_r,m)';
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27
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28 switch order
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29
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30 case 6
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31
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32 [obj.H, obj.HI, obj.D1, D2, ...
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33 ~, obj.e_l, obj.e_r, ~, ~, ~, ~, ~,...
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34 d1_l, d1_r] = ...
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35 sbp.implementations.d4_variable_6(m, obj.h);
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36
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37 case 4
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38 [obj.H, obj.HI, obj.D1, D2, obj.e_l,...
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39 obj.e_r, d1_l, d1_r] = ...
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40 sbp.implementations.d2_variable_4(m,obj.h);
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41 case 2
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42 [obj.H, obj.HI, obj.D1, D2, obj.e_l,...
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43 obj.e_r, d1_l, d1_r] = ...
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44 sbp.implementations.d2_variable_2(m,obj.h);
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45
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46 otherwise
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47 error('Invalid operator order %d.',order);
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48 end
2924b3a9b921 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.
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49 obj.borrowing.H11 = obj.H(1,1)/obj.h; % First element in H/h,
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50 obj.borrowing.M.d1 = obj.H(1,1)/obj.h; % First element in H/h is borrowing also for M
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51 obj.borrowing.R.delta_D = inf; % Because delta_D is zero, one can borrow infinitely much.
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52 % This sets penalties of the form 1/borrowing to 0, which is
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53 % the desired behaviour.
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54 obj.m = m;
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55 obj.M = [];
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56
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57 D1 = obj.D1;
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58 e_r = obj.e_r;
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59 e_l = obj.e_l;
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60
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61 % D2 = Hinv * (-M + br*er*d1r^T - bl*el*d1l^T);
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62 % Replace d1' by e'*D1 in D2.
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63 D2_compatible = @(b) D2(b) - obj.HI*(b(m)*e_r*d1_r' - b(m)*e_r*e_r'*D1) ...
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64 + obj.HI*(b(1)*e_l*d1_l' - b(1)*e_l*e_l'*D1);
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65
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66 obj.D2 = D2_compatible;
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67 obj.d1_l = (e_l'*D1)';
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68 obj.d1_r = (e_r'*D1)';
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69
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70 end
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71 function str = string(obj)
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72 str = [class(obj) '_' num2str(obj.order)];
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73 end
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74 end
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76 end
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81