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annotate diracDiscr.m @ 1242:ff613067dec6 feature/dirac_discr

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author Martin Almquist <malmquist@stanford.edu>
date Tue, 19 Nov 2019 17:04:32 -0800
parents d1b201fe328e a0940c1db455
children 0a1c64d3c717
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2 function d = diracDiscr(g, x_s, m_order, s_order, H)
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3 % n-dimensional delta function
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4 % g: cartesian grid
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5 % x_s: source point coordinate vector, e.g. [x; y] or [x; y; z].
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6 % m_order: Number of moment conditions
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7 % s_order: Number of smoothness conditions
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8 % H: cell array of 1D norm matrices
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9 assertType(g, 'grid.Cartesian');
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10 dim = g.d;
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11 d_1D = cell(dim,1);
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13 % Allow for non-cell input in 1D
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14 if dim == 1
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15 H = {H};
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16 end
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17 % Create 1D dirac discr for each coordinate dir.
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18 for i = 1:dim
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19 d_1D{i} = diracDiscr1D(x_s(i), g.x{i}, m_order, s_order, H{i});
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20 end
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22 d = d_1D{dim};
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23 for i = dim-1: -1: 1
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24 % Perform outer product, transpose, and then turn into column vector
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25 d = (d_1D{i}*d')';
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26 d = d(:);
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27 end
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29 end
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32 % Helper function for 1D delta functions
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33 function ret = diracDiscr1D(x_s, x, m_order, s_order, H)
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35 % Return zeros if x0 is outside grid
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36 if x_s < x(1) || x_s > x(end)
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37 ret = zeros(size(x));
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38 return
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39 else
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40 tot_order = m_order+s_order; %This is equiv. to the number of equations solved for
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41 S = [];
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42 M = [];
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44 % Get interior grid spacing
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45 middle = floor(length(x)/2);
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46 h = x(middle+1) - x(middle); % Use middle point to allow for staggered grids.
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48 index = sourceIndices(x_s, x, tot_order, h);
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50 polynomial = (x(index)-x(index(1)))/(x(index(end))-x(index(1)));
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51 x_0 = (x_s-x(index(1)))/(x(index(end))-x(index(1)));
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52
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53 quadrature = diag(H);
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54 quadrature_weights = quadrature(index)/h;
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56 h_polynomial = polynomial(2)-polynomial(1);
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57 b = zeros(tot_order,1);
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59 for i = 1:m_order
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60 b(i,1) = x_0^(i-1);
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61 end
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63 for i = 1:tot_order
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64 for j = 1:m_order
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65 M(j,i) = polynomial(i)^(j-1)*h_polynomial*quadrature_weights(i);
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66 end
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67 end
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69 for i = 1:tot_order
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70 for j = 1:s_order
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71 S(j,i) = (-1)^(i-1)*polynomial(i)^(j-1);
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72 end
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73 end
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74
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75 A = [M;S];
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76
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77 d = A\b;
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78 ret = x*0;
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79 ret(index) = d/h*h_polynomial;
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80 end
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82 end
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85 function I = sourceIndices(x_s, x, tot_order, h)
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86 % Find the indices that are within range of of the point source location
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87 I = find(tot_order*h/2 >= abs(x-x_s));
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88
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89 if length(I) > tot_order
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90 if length(I) == tot_order + 2
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91 I = I(2:end-1);
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92 elseif length(I) == tot_order + 1
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93 I = I(1:end-1);
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94 end
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95 elseif length(I) < tot_order
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96 if x_s < x(1) + ceil(tot_order/2)*h
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97 I = 1:tot_order;
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98 elseif x_s > x(end) - ceil(tot_order/2)*h
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99 I = length(x)-tot_order+1:length(x);
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100 else
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101 if I(end) < length(x)
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102 I = [I; I(end)+1];
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103 else
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104 I = [I(1)-1; I];
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105 end
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106 end
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107 end
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108 end