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1 classdef SchmWave < 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 D2
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14 Hi
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15 e_l
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16 e_r
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17 d1_l
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18 d1_r
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19 gamm
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20 end
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21
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22 methods
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23 function obj = SchmWave(m,xlim,order,alpha)
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24 default_arg('a',1);
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25 [x, h] = util.get_grid(xlim{:},m);
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26
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27 ops = sbp.Ordinary(m,h,order);
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28
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29 obj.D2 = sparse(ops.derivatives.D2);
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30 obj.H = sparse(ops.norms.H);
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31 obj.Hi = sparse(ops.norms.HI);
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32 obj.M = sparse(ops.norms.M);
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33 obj.e_l = sparse(ops.boundary.e_1);
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34 obj.e_r = sparse(ops.boundary.e_m);
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35 obj.d1_l = sparse(ops.boundary.S_1);
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36 obj.d1_r = sparse(ops.boundary.S_m);
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37
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38
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39 obj.m = m;
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40 obj.h = h;
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41 obj.order = order;
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42
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43 obj.alpha = alpha;
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44 obj.D = alpha*obj.D2;
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45 obj.x = x;
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46
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47 obj.gamm = h*ops.borrowing.M.S;
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48
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49 end
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50
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51
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52 % Closure functions return the opertors applied to the own doamin to close the boundary
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53 % 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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54 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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55 % type is a string specifying the type of boundary condition if there are several.
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56 % data is a function returning the data that should be applied at the boundary.
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57 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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58 % neighbour_boundary is a string specifying which boundary to interface to.
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59 function [closure, penalty] = boundary_condition(obj,boundary,type,data)
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60 default_arg('type','neumann');
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61 default_arg('data',0);
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62
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63 [e,d,s] = obj.get_boundary_ops(boundary);
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64
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65 switch type
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66 % Dirichlet boundary condition
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67 case {'D','dirichlet'}
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68 alpha = obj.alpha;
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69
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70 % tau1 < -alpha^2/gamma
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71 tuning = 1.1;
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72 tau1 = -tuning*alpha/obj.gamm;
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73 tau2 = s*alpha;
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74
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75 p = tau1*e + tau2*d;
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76
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77 closure = obj.Hi*p*e';
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78
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79 pp = obj.Hi*p;
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80 switch class(data)
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81 case 'double'
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82 penalty = pp*data;
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83 case 'function_handle'
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84 penalty = @(t)pp*data(t);
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85 otherwise
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86 error('Wierd data argument!')
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87 end
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88
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89
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90 % Neumann boundary condition
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91 case {'N','neumann'}
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92 alpha = obj.alpha;
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93 tau1 = -s*alpha;
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94 tau2 = 0;
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95 tau = tau1*e + tau2*d;
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96
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97 closure = obj.Hi*tau*d';
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98
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99 pp = obj.Hi*tau;
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100 switch class(data)
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101 case 'double'
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102 penalty = pp*data;
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103 case 'function_handle'
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104 penalty = @(t)pp*data(t);
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105 otherwise
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106 error('Wierd data argument!')
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107 end
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108
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109 % Unknown, boundary condition
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110 otherwise
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111 error('No such boundary condition: type = %s',type);
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112 end
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113 end
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114
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115 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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116 % u denotes the solution in the own domain
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117 % v denotes the solution in the neighbour domain
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118 [e_u,d_u,s_u] = obj.get_boundary_ops(boundary);
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119 [e_v,d_v,s_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
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120
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121 tuning = 1.1;
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122
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123 alpha_u = obj.alpha;
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124 alpha_v = neighbour_scheme.alpha;
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125
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126 gamm_u = obj.gamm;
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127 gamm_v = neighbour_scheme.gamm;
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128
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129 % tau1 < -(alpha_u/gamm_u + alpha_v/gamm_v)
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130
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131 tau1 = -(alpha_u/gamm_u + alpha_v/gamm_v) * tuning;
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132 tau2 = s_u*1/2*alpha_u;
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133 sig1 = s_u*(-1/2);
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134 sig2 = 0;
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135
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136 tau = tau1*e_u + tau2*d_u;
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137 sig = sig1*e_u + sig2*d_u;
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138
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139 closure = obj.Hi*( tau*e_u' + sig*alpha_u*d_u');
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140 penalty = obj.Hi*(-tau*e_v' - sig*alpha_v*d_v');
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141 end
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142
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143 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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144 % The right boundary is considered the positive boundary
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145 function [e,d,s] = get_boundary_ops(obj,boundary)
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146 switch boundary
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147 case 'l'
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148 e = obj.e_l;
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149 d = obj.d1_l;
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150 s = -1;
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151 case 'r'
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152 e = obj.e_r;
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153 d = obj.d1_r;
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154 s = 1;
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155 otherwise
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156 error('No such boundary: boundary = %s',boundary);
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157 end
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158 end
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159
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160 function N = size(obj)
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161 N = obj.m;
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162 end
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163
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164 end
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165
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166 methods(Static)
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167 % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u
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168 % and bound_v of scheme schm_v.
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169 % [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l')
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170 function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v)
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171 [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v);
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172 [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u);
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173 end
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174 end
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175 end |
