Mercurial > repos > public > sbplib
annotate +scheme/Schrodinger.m @ 1198:2924b3a9b921 feature/d2_compatible
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 | 0c504a21432d |
| children |
| rev | line source |
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1 classdef Schrodinger < scheme.Scheme |
| 0 | 2 properties |
| 3 m % Number of points in each direction, possibly a vector | |
| 4 h % Grid spacing | |
| 5 x % Grid | |
| 6 order % Order accuracy for the approximation | |
| 7 | |
| 8 D % non-stabalized scheme operator | |
| 9 H % Discrete norm | |
| 10 M % Derivative norm | |
| 11 alpha | |
| 12 | |
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13 D2 |
| 0 | 14 Hi |
| 15 e_l | |
| 16 e_r | |
| 17 d1_l | |
| 18 d1_r | |
| 19 gamm | |
| 20 end | |
| 21 | |
| 22 methods | |
| 23 % Solving SE in the form u_t = i*u_xx -i*V; | |
| 24 function obj = Schrodinger(m,xlim,order,V) | |
| 25 default_arg('V',0); | |
| 26 | |
| 27 [x, h] = util.get_grid(xlim{:},m); | |
| 28 | |
| 29 ops = sbp.Ordinary(m,h,order); | |
| 30 | |
| 31 obj.D2 = sparse(ops.derivatives.D2); | |
| 32 obj.H = sparse(ops.norms.H); | |
| 33 obj.Hi = sparse(ops.norms.HI); | |
| 34 obj.M = sparse(ops.norms.M); | |
| 35 obj.e_l = sparse(ops.boundary.e_1); | |
| 36 obj.e_r = sparse(ops.boundary.e_m); | |
| 37 obj.d1_l = sparse(ops.boundary.S_1); | |
| 38 obj.d1_r = sparse(ops.boundary.S_m); | |
| 39 | |
| 40 | |
| 41 if isa(V,'function_handle') | |
| 42 V_vec = V(x); | |
| 43 else | |
| 44 V_vec = x*0 + V; | |
| 45 end | |
| 46 | |
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47 V_mat = spdiags(V_vec,0,m,m); |
| 0 | 48 |
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49 obj.D = 1i * obj.D2 - 1i * V_mat; |
| 0 | 50 |
| 51 obj.m = m; | |
| 52 obj.h = h; | |
| 53 obj.order = order; | |
| 54 | |
| 55 obj.x = x; | |
| 56 end | |
| 57 | |
| 58 | |
| 59 % Closure functions return the opertors applied to the own doamin to close the boundary | |
| 60 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin. | |
| 61 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'. | |
| 62 % type is a string specifying the type of boundary condition if there are several. | |
| 63 % data is a function returning the data that should be applied at the boundary. | |
| 64 % neighbour_scheme is an instance of Scheme that should be interfaced to. | |
| 65 % neighbour_boundary is a string specifying which boundary to interface to. | |
| 66 function [closure, penalty] = boundary_condition(obj,boundary,type,data) | |
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67 default_arg('type','dirichlet'); |
| 0 | 68 default_arg('data',0); |
| 69 | |
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70 [e, d] = obj.getBoundaryOperator({'e', 'd'}, boundary); |
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71 s = obj.getBoundarySign(boundary); |
| 0 | 72 |
| 73 switch type | |
| 74 % Dirichlet boundary condition | |
| 75 case {'D','d','dirichlet'} | |
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76 tau = s * 1i*d; |
| 0 | 77 closure = obj.Hi*tau*e'; |
| 78 | |
| 79 switch class(data) | |
| 80 case 'double' | |
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81 penalty = -obj.Hi*tau*data; |
| 0 | 82 case 'function_handle' |
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83 penalty = @(t)-obj.Hi*tau*data(t); |
| 0 | 84 otherwise |
| 85 error('Wierd data argument!') | |
| 86 end | |
| 87 | |
| 88 % Unknown, boundary condition | |
| 89 otherwise | |
| 90 error('No such boundary condition: type = %s',type); | |
| 91 end | |
| 92 end | |
| 93 | |
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94 function [closure, penalty] = interface(obj, boundary, neighbour_scheme, neighbour_boundary, type) |
| 0 | 95 % u denotes the solution in the own domain |
| 96 % v denotes the solution in the neighbour domain | |
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97 [e_u, d_u] = obj.getBoundaryOperator({'e', 'd'}, boundary); |
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98 s_u = obj.getBoundarySign(boundary); |
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99 |
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100 [e_v, d_v] = neighbour_scheme.getBoundaryOperator({'e', 'd'}, neighbour_boundary); |
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101 s_v = neighbour_scheme.getBoundarySign(neighbour_boundary); |
| 0 | 102 |
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103 a = -s_u* 1/2 * 1i ; |
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104 b = a'; |
| 0 | 105 |
| 106 tau = b*d_u; | |
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107 sig = -a*e_u; |
| 0 | 108 |
| 109 closure = obj.Hi * (tau*e_u' + sig*d_u'); | |
| 110 penalty = obj.Hi * (-tau*e_v' - sig*d_v'); | |
| 111 end | |
| 112 | |
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113 % Returns the boundary operator op for the boundary specified by the string boundary. |
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114 % op -- string or a cell array of strings |
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115 % boundary -- string |
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116 function varargout = getBoundaryOperator(obj, op, boundary) |
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117 assertIsMember(boundary, {'l', 'r'}) |
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118 |
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119 if ~iscell(op) |
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120 op = {op}; |
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121 end |
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122 |
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123 for i = 1:numel(op) |
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124 switch op{i} |
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125 case 'e' |
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126 switch boundary |
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127 case 'l' |
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128 e = obj.e_l; |
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129 case 'r' |
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130 e = obj.e_r; |
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131 end |
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132 varargout{i} = e; |
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133 |
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134 case 'd' |
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135 switch boundary |
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136 case 'l' |
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137 d = obj.d1_l; |
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138 case 'r' |
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139 d = obj.d1_r; |
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140 end |
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141 varargout{i} = d; |
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142 end |
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143 end |
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144 end |
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145 |
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146 % Returns square boundary quadrature matrix, of dimension |
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147 % corresponding to the number of boundary points |
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148 % |
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149 % boundary -- string |
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150 % Note: for 1d diffOps, the boundary quadrature is the scalar 1. |
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151 function H_b = getBoundaryQuadrature(obj, boundary) |
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152 assertIsMember(boundary, {'l', 'r'}) |
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153 |
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154 H_b = 1; |
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155 end |
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156 |
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157 % Returns the boundary sign. The right boundary is considered the positive boundary |
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158 % boundary -- string |
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159 function s = getBoundarySign(obj, boundary) |
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160 assertIsMember(boundary, {'l', 'r'}) |
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161 |
| 0 | 162 switch boundary |
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163 case {'r'} |
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164 s = 1; |
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165 case {'l'} |
| 0 | 166 s = -1; |
| 167 end | |
| 168 end | |
| 169 | |
| 170 function N = size(obj) | |
| 171 N = obj.m; | |
| 172 end | |
| 173 | |
| 174 end | |
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175 end |