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1 classdef Schrodinger < noname.Scheme
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2 properties
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3 m % Number of points in each direction, possibly a vector
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4 h % Grid spacing
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5 x % Grid
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6 order % Order accuracy for the approximation
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7
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8 D % non-stabalized scheme operator
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9 H % Discrete norm
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10 M % Derivative norm
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11 alpha
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12
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13 Hi
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14 e_l
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15 e_r
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16 d1_l
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17 d1_r
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18 gamm
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19 end
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20
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21 methods
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22 % Solving SE in the form u_t = i*u_xx -i*V;
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23 function obj = Schrodinger(m,xlim,order,V)
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24 default_arg('V',0);
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25
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26 [x, h] = util.get_grid(xlim{:},m);
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27
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28 ops = sbp.Ordinary(m,h,order);
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29
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30 obj.D2 = sparse(ops.derivatives.D2);
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31 obj.H = sparse(ops.norms.H);
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32 obj.Hi = sparse(ops.norms.HI);
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33 obj.M = sparse(ops.norms.M);
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34 obj.e_l = sparse(ops.boundary.e_1);
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35 obj.e_r = sparse(ops.boundary.e_m);
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36 obj.d1_l = sparse(ops.boundary.S_1);
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37 obj.d1_r = sparse(ops.boundary.S_m);
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38
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39
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40 if isa(V,'function_handle')
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41 V_vec = V(x);
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42 else
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43 V_vec = x*0 + V;
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44 end
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45
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46 V_mat = spdiags(V,0,m,m);
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47
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48
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49 D = 1i * obj.D2 - 1i * V;
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50
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51 obj.m = m;
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52 obj.h = h;
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53 obj.order = order;
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54
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55 obj.D = alpha*obj.D2;
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56 obj.x = x;
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57 end
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58
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59
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60 % Closure functions return the opertors applied to the own doamin to close the boundary
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61 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
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62 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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63 % type is a string specifying the type of boundary condition if there are several.
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64 % data is a function returning the data that should be applied at the boundary.
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65 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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66 % neighbour_boundary is a string specifying which boundary to interface to.
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67 function [closure, penalty] = boundary_condition(obj,boundary,type,data)
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68 default_arg('type','neumann');
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69 default_arg('data',0);
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70
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71 [e,d,s] = obj.get_boundary_ops(boundary);
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72
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73 switch type
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74 % Dirichlet boundary condition
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75 case {'D','d','dirichlet'}
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76 tau = -1i* s * d;
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77
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78 closure = obj.Hi*tau*e';
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79
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80 pp = obj.Hi*p;
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81 switch class(data)
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82 case 'double'
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83 penalty = pp*data;
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84 case 'function_handle'
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85 penalty = @(t)pp*data(t);
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86 otherwise
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87 error('Wierd data argument!')
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88 end
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89
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90 % Unknown, boundary condition
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91 otherwise
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92 error('No such boundary condition: type = %s',type);
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93 end
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94 end
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95
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96 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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97 % u denotes the solution in the own domain
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98 % v denotes the solution in the neighbour domain
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99 [e_u,d_u,s_u] = obj.get_boundary_ops(boundary);
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100 [e_v,d_v,s_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
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101
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102 a = s/2 * 1i ;
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103 b = - a';
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104
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105 tau = b*d_u;
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106 sig = a*e_u;
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107
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108 closure = obj.Hi * (tau*e_u' + sig*d_u');
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109 penalty = obj.Hi * (-tau*e_v' - sig*d_v');
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110 end
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111
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112 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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113 % The right boundary is considered the positive boundary
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114 function [e,d,s] = get_boundary_ops(obj,boundary)
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115 switch boundary
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116 case 'l'
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117 e = obj.e_l;
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118 d = obj.d1_l;
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119 s = -1;
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120 case 'r'
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121 e = obj.e_r;
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122 d = obj.d1_r;
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123 s = 1;
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124 otherwise
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125 error('No such boundary: boundary = %s',boundary);
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126 end
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127 end
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128
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129 function N = size(obj)
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130 N = obj.m;
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131 end
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132
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133 end
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134
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135 methods(Static)
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136 % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u
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137 % and bound_v of scheme schm_v.
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138 % [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l')
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139 function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v)
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140 [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v);
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141 [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u);
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142 end
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143 end
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144 end |
