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Refactor interface method for Schrodinger2d
author Martin Almquist <malmquist@stanford.edu>
date Sun, 25 Nov 2018 19:46:39 -0800
parents b9c98661ff5d
children 46f5dc61d90b
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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1 classdef Schrodinger2d < scheme.Scheme
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2
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3 % Discretizes the Laplacian with constant coefficent,
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4 % in the Schrödinger equation way (i.e., the discretization matrix is not necessarily
718
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5 % definite)
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6 % u_t = a*i*Laplace u
718
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7 % opSet should be cell array of opSets, one per dimension. This
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8 % is useful if we have periodic BC in one direction.
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9
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10 properties
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11 m % Number of points in each direction, possibly a vector
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12 h % Grid spacing
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13
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14 grid
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15 dim
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16
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17 order % Order of accuracy for the approximation
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18
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19 % Diagonal matrix for variable coefficients
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20 a % Constant coefficient
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21
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22 D % Total operator
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23 D1 % First derivatives
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24
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25 % Second derivatives
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26 D2
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27
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28 H, Hi % Inner products
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29 e_l, e_r
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30 d1_l, d1_r % Normal derivatives at the boundary
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31 e_w, e_e, e_s, e_n
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32 d_w, d_e, d_s, d_n
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33
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34 H_boundary % Boundary inner products
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35
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36 end
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37
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38 methods
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39
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1cdf5ead2a16 Refactor interface method for Schrodinger2d
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40 function obj = Schrodinger2d(g ,order, a, opSet)
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41 default_arg('opSet',{@sbp.D2Variable, @sbp.D2Variable});
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42 default_arg('a',1);
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43 dim = 2;
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44
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45 assert(isa(g, 'grid.Cartesian'))
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46 if isa(a, 'function_handle')
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47 a = grid.evalOn(g, a);
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48 a = spdiag(a);
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49 end
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50
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51 m = g.size();
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52 m_tot = g.N();
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53
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54 h = g.scaling();
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55 xlim = {g.x{1}(1), g.x{1}(end)};
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56 ylim = {g.x{2}(1), g.x{2}(end)};
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57 lim = {xlim, ylim};
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58
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59 % 1D operators
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60 ops = cell(dim,1);
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61 for i = 1:dim
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62 ops{i} = opSet{i}(m(i), lim{i}, order);
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63 end
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64
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65 I = cell(dim,1);
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66 D1 = cell(dim,1);
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67 D2 = cell(dim,1);
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68 H = cell(dim,1);
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69 Hi = cell(dim,1);
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70 e_l = cell(dim,1);
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71 e_r = cell(dim,1);
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72 d1_l = cell(dim,1);
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73 d1_r = cell(dim,1);
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74
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75 for i = 1:dim
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76 I{i} = speye(m(i));
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77 D1{i} = ops{i}.D1;
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78 D2{i} = ops{i}.D2;
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79 H{i} = ops{i}.H;
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80 Hi{i} = ops{i}.HI;
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81 e_l{i} = ops{i}.e_l;
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82 e_r{i} = ops{i}.e_r;
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83 d1_l{i} = ops{i}.d1_l;
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84 d1_r{i} = ops{i}.d1_r;
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85 end
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86
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87 % Constant coeff D2
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88 for i = 1:dim
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89 D2{i} = D2{i}(ones(m(i),1));
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90 end
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91
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92 %====== Assemble full operators ========
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93 obj.D1 = cell(dim,1);
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94 obj.D2 = cell(dim,1);
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95 obj.e_l = cell(dim,1);
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96 obj.e_r = cell(dim,1);
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97 obj.d1_l = cell(dim,1);
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98 obj.d1_r = cell(dim,1);
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99
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100 % D1
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101 obj.D1{1} = kron(D1{1},I{2});
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102 obj.D1{2} = kron(I{1},D1{2});
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103
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104 % Boundary operators
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105 obj.e_l{1} = kron(e_l{1},I{2});
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106 obj.e_l{2} = kron(I{1},e_l{2});
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107 obj.e_r{1} = kron(e_r{1},I{2});
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108 obj.e_r{2} = kron(I{1},e_r{2});
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109
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110 obj.d1_l{1} = kron(d1_l{1},I{2});
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111 obj.d1_l{2} = kron(I{1},d1_l{2});
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112 obj.d1_r{1} = kron(d1_r{1},I{2});
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113 obj.d1_r{2} = kron(I{1},d1_r{2});
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114
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115 % D2
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116 obj.D2{1} = kron(D2{1},I{2});
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117 obj.D2{2} = kron(I{1},D2{2});
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118
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119 % Quadratures
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120 obj.H = kron(H{1},H{2});
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121 obj.Hi = inv(obj.H);
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122 obj.H_boundary = cell(dim,1);
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123 obj.H_boundary{1} = H{2};
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124 obj.H_boundary{2} = H{1};
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125
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126 % Differentiation matrix D (without SAT)
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127 D2 = obj.D2;
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128 D = sparse(m_tot,m_tot);
71aa5828cbbf Add Schrödinger scheme for 2d single block. Will develop to multiblock with interpolation.
Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
129 for j = 1:dim
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
130 D = D + a*1i*D2{j};
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
131 end
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
132 obj.D = D;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
133 %=========================================%
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
134
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
135 % Misc.
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
136 obj.m = m;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
137 obj.h = h;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
138 obj.order = order;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
139 obj.grid = g;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
140 obj.dim = dim;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
141 obj.a = a;
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
142 obj.e_w = obj.e_l{1};
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
143 obj.e_e = obj.e_r{1};
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
144 obj.e_s = obj.e_l{2};
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
145 obj.e_n = obj.e_r{2};
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
146 obj.d_w = obj.d1_l{1};
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Martin Almquist <malmquist@stanford.edu>
parents: 718
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147 obj.d_e = obj.d1_r{1};
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Martin Almquist <malmquist@stanford.edu>
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diff changeset
148 obj.d_s = obj.d1_l{2};
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diff changeset
149 obj.d_n = obj.d1_r{2};
718
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parents:
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150
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Martin Almquist <malmquist@stanford.edu>
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diff changeset
151 end
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152
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153
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Martin Almquist <malmquist@stanford.edu>
parents:
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154 % Closure functions return the operators applied to the own domain to close the boundary
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155 % Penalty functions return the operators to force the solution. In the case of an interface it returns the operator applied to the other doamin.
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parents:
diff changeset
156 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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157 % type is a string specifying the type of boundary condition.
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158 % data is a function returning the data that should be applied at the boundary.
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parents:
diff changeset
159 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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parents:
diff changeset
160 % neighbour_boundary is a string specifying which boundary to interface to.
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parents:
diff changeset
161 function [closure, penalty] = boundary_condition(obj, boundary, type, parameter)
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
162 default_arg('type','Neumann');
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
163 default_arg('parameter', []);
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
164
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
165 % j is the coordinate direction of the boundary
905
459eeb99130f Include type as (optional) input parameter in the interface method of all schemes.
Martin Almquist <malmquist@stanford.edu>
parents: 743
diff changeset
166 % nj: outward unit normal component.
718
71aa5828cbbf Add Schrödinger scheme for 2d single block. Will develop to multiblock with interpolation.
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parents:
diff changeset
167 % nj = -1 for west, south, bottom boundaries
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parents:
diff changeset
168 % nj = 1 for east, north, top boundaries
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parents:
diff changeset
169 [j, nj] = obj.get_boundary_number(boundary);
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
170 switch nj
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
171 case 1
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
172 e = obj.e_r;
71aa5828cbbf Add Schrödinger scheme for 2d single block. Will develop to multiblock with interpolation.
Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
173 d = obj.d1_r;
71aa5828cbbf Add Schrödinger scheme for 2d single block. Will develop to multiblock with interpolation.
Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
174 case -1
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
175 e = obj.e_l;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
176 d = obj.d1_l;
71aa5828cbbf Add Schrödinger scheme for 2d single block. Will develop to multiblock with interpolation.
Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
177 end
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diff changeset
178
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
179 Hi = obj.Hi;
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
180 H_gamma = obj.H_boundary{j};
743
f4595f14d696 Change schemes to work for special coefficients.
Martin Almquist <malmquist@stanford.edu>
parents: 719
diff changeset
181 a = e{j}'*obj.a*e{j};
718
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parents:
diff changeset
182
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
183 switch type
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Martin Almquist <malmquist@stanford.edu>
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diff changeset
184
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
185 % Dirichlet boundary condition
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parents:
diff changeset
186 case {'D','d','dirichlet','Dirichlet'}
905
459eeb99130f Include type as (optional) input parameter in the interface method of all schemes.
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parents: 743
diff changeset
187 closure = nj*Hi*d{j}*a*1i*H_gamma*(e{j}' );
718
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188 penalty = -nj*Hi*d{j}*a*1i*H_gamma;
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189
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
190 % Free boundary condition
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parents:
diff changeset
191 case {'N','n','neumann','Neumann'}
905
459eeb99130f Include type as (optional) input parameter in the interface method of all schemes.
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diff changeset
192 closure = -nj*Hi*e{j}*a*1i*H_gamma*(d{j}' );
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diff changeset
193 penalty = nj*Hi*e{j}*a*1i*H_gamma;
718
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194
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
195 % Unknown boundary condition
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
196 otherwise
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
197 error('No such boundary condition: type = %s',type);
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diff changeset
198 end
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Martin Almquist <malmquist@stanford.edu>
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199 end
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200
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
201 % opts Struct that specifies the interface coupling.
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Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
202 % Fields:
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
203 % -- interpolation: struct of interpolation operators (empty for conforming grids)
910
b9c98661ff5d Change the name of the last interface method argument from type to opts in all schemes.
Martin Almquist <malmquist@stanford.edu>
parents: 905
diff changeset
204 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary,opts)
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
205 if isempty(opts)
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parents: 910
diff changeset
206 [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary);
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
207 else
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Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
208 assertType(opts, 'struct');
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
209 if isfield(opts, 'I_local2neighbor')
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Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
210 [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,opts);
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
211 else
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
212 [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,opts);
1cdf5ead2a16 Refactor interface method for Schrodinger2d
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parents: 910
diff changeset
213 end
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
214 end
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
215 end
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
216
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
217 function [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,opts)
718
71aa5828cbbf Add Schrödinger scheme for 2d single block. Will develop to multiblock with interpolation.
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parents:
diff changeset
218 % u denotes the solution in the own domain
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Martin Almquist <malmquist@stanford.edu>
parents:
diff changeset
219 % v denotes the solution in the neighbour domain
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
220
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
221 % Get boundary operators
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
222 [e_neighbour, d_neighbour] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
1cdf5ead2a16 Refactor interface method for Schrodinger2d
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parents: 910
diff changeset
223 [e, d, H_gamma] = obj.get_boundary_ops(boundary);
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
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parents: 718
diff changeset
224 Hi = obj.Hi;
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
225 a = obj.a;
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diff changeset
226
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
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diff changeset
227 % Get outward unit normal component
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parents: 910
diff changeset
228 [~, n] = obj.get_boundary_number(boundary);
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
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parents: 718
diff changeset
229
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
230 Hi = obj.Hi;
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Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
231 sigma = -n*1i*a/2;
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
232 tau = -n*(1i*a)'/2;
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
233
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
234 closure = tau*Hi*d*H_gamma*e' + sigma*Hi*e*H_gamma*d';
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
235 penalty = -tau*Hi*d*H_gamma*e_neighbour' ...
1cdf5ead2a16 Refactor interface method for Schrodinger2d
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parents: 910
diff changeset
236 -sigma*Hi*e*H_gamma*d_neighbour';
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
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parents: 718
diff changeset
237
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
238 end
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
239
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
240 function [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,opts)
1cdf5ead2a16 Refactor interface method for Schrodinger2d
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parents: 910
diff changeset
241 % u denotes the solution in the own domain
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
242 % v denotes the solution in the neighbour domain
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
Martin Almquist <malmquist@stanford.edu>
parents: 718
diff changeset
243
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
244 % Get boundary operators
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
245 [e_neighbour, d_neighbour] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
246 [e, d, H_gamma] = obj.get_boundary_ops(boundary);
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
247 Hi = obj.Hi;
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
248 a = obj.a;
1cdf5ead2a16 Refactor interface method for Schrodinger2d
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parents: 910
diff changeset
249
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
250 % Get outward unit normal component
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
251 [~, n] = obj.get_boundary_number(boundary);
719
b3f8fb9cefd2 Add interpolation to Schrödinger 2D scheme.
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parents: 718
diff changeset
252
912
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
253 % Get interpolation operators from opts
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
254 I_u2v_good = opts.I_local2neighbor.good;
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
255 I_u2v_bad = opts.I_local2neighbor.bad;
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
256 I_v2u_good = opts.I_neighbor2local.good;
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
257 I_v2u_bad = opts.I_neighbor2local.bad;
1cdf5ead2a16 Refactor interface method for Schrodinger2d
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parents: 910
diff changeset
258
1cdf5ead2a16 Refactor interface method for Schrodinger2d
Martin Almquist <malmquist@stanford.edu>
parents: 910
diff changeset
259 sigma = -n*1i*a/2;
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260 tau = -n*(1i*a)'/2;
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261
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262 closure = tau*Hi*d*H_gamma*e' + sigma*Hi*e*H_gamma*d';
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263 penalty = -tau*Hi*d*H_gamma*I_v2u_good*e_neighbour' ...
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264 -sigma*Hi*e*H_gamma*I_v2u_bad*d_neighbour';
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265
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266 end
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267
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268 % Returns the coordinate number and outward normal component for the boundary specified by the string boundary.
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269 function [j, nj] = get_boundary_number(obj, boundary)
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270
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271 switch boundary
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272 case {'w','W','west','West', 'e', 'E', 'east', 'East'}
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273 j = 1;
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274 case {'s','S','south','South', 'n', 'N', 'north', 'North'}
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275 j = 2;
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276 otherwise
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277 error('No such boundary: boundary = %s',boundary);
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278 end
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279
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280 switch boundary
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281 case {'w','W','west','West','s','S','south','South'}
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282 nj = -1;
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283 case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
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284 nj = 1;
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285 end
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286 end
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287
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288 % Returns the boundary ops and sign for the boundary specified by the string boundary.
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289 % The right boundary is considered the positive boundary
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290 function [e, d, H_b] = get_boundary_ops(obj, boundary)
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291
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292 switch boundary
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293 case 'w'
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294 e = obj.e_w;
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295 d = obj.d_w;
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296 H_b = obj.H_boundary{1};
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297 case 'e'
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298 e = obj.e_e;
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299 d = obj.d_e;
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300 H_b = obj.H_boundary{1};
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301 case 's'
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302 e = obj.e_s;
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303 d = obj.d_s;
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304 H_b = obj.H_boundary{2};
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305 case 'n'
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306 e = obj.e_n;
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307 d = obj.d_n;
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308 H_b = obj.H_boundary{2};
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309 otherwise
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310 error('No such boundary: boundary = %s',boundary);
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311 end
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312 end
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313
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314 function N = size(obj)
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315 N = prod(obj.m);
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316 end
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317 end
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318 end