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1 classdef SchmWave2d < 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,y % Grid
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6 X,Y % Values of x and y for each grid point
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7 order % Order accuracy for the approximation
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8
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9 D % non-stabalized scheme operator
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10 M % Derivative norm
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11 alpha
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12
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13 H % Discrete norm
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14 Hi
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15 H_x, H_y % Norms in the x and y directions
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16 Hx,Hy % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
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17 Hi_x, Hi_y
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18 Hix, Hiy
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19 e_w, e_e, e_s, e_n
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20 d1_w, d1_e, d1_s, d1_n
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21 gamm_x, gamm_y
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22 end
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23
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24 methods
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25 function obj = SchmWave2d(m,xlim,ylim,order,alpha)
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26 default_arg('a',1);
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27
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28 if length(m) == 1
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29 m = [m m];
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30 end
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31
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32 m_x = m(1);
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33 m_y = m(2);
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34
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35 [x, h_x] = util.get_grid(xlim{:},m_x);
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36 [y, h_y] = util.get_grid(ylim{:},m_y);
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37
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38 ops_x = sbp.Ordinary(m_x,h_x,order);
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39 ops_y = sbp.Ordinary(m_y,h_y,order);
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40
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41 I_x = speye(m_x);
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42 I_y = speye(m_y);
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43
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44 D2_x = sparse(ops_x.derivatives.D2);
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45 H_x = sparse(ops_x.norms.H);
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46 Hi_x = sparse(ops_x.norms.HI);
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47 M_x = sparse(ops_x.norms.M);
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48 e_l_x = sparse(ops_x.boundary.e_1);
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49 e_r_x = sparse(ops_x.boundary.e_m);
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50 d1_l_x = sparse(ops_x.boundary.S_1);
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51 d1_r_x = sparse(ops_x.boundary.S_m);
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52
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53 D2_y = sparse(ops_y.derivatives.D2);
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54 H_y = sparse(ops_y.norms.H);
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55 Hi_y = sparse(ops_y.norms.HI);
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56 M_y = sparse(ops_y.norms.M);
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57 e_l_y = sparse(ops_y.boundary.e_1);
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58 e_r_y = sparse(ops_y.boundary.e_m);
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59 d1_l_y = sparse(ops_y.boundary.S_1);
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60 d1_r_y = sparse(ops_y.boundary.S_m);
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61
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62 D2 = kr(D2_x, I_y) + kr(I_x, D2_y);
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63 obj.H = kr(H_x,H_y);
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64 obj.Hx = kr(H_x,I_y);
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65 obj.Hy = kr(I_x,H_y);
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66 obj.Hix = kr(Hi_x,I_y);
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67 obj.Hiy = kr(I_x,Hi_y);
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68 obj.Hi = kr(Hi_x,Hi_y);
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69 obj.M = kr(M_x,H_y)+kr(H_x,M_y);
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70 obj.e_w = kr(e_l_x,I_y);
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71 obj.e_e = kr(e_r_x,I_y);
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72 obj.e_s = kr(I_x,e_l_y);
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73 obj.e_n = kr(I_x,e_r_y);
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74 obj.d1_w = kr(d1_l_x,I_y);
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75 obj.d1_e = kr(d1_r_x,I_y);
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76 obj.d1_s = kr(I_x,d1_l_y);
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77 obj.d1_n = kr(I_x,d1_r_y);
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78
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79 obj.m = m;
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80 obj.h = [h_x h_y];
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81 obj.order = order;
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82
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83 obj.alpha = alpha;
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84 obj.D = alpha*D2;
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85 obj.x = x;
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86 obj.y = y;
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87 obj.X = kr(x,ones(m_y,1));
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88 obj.Y = kr(ones(m_x,1),y);
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89
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90 obj.gamm_x = h_x*ops_x.borrowing.M.S;
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91 obj.gamm_y = h_y*ops_y.borrowing.M.S;
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92 end
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93
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94
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95 % Closure functions return the opertors applied to the own doamin to close the boundary
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96 % 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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97 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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98 % type is a string specifying the type of boundary condition if there are several.
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99 % data is a function returning the data that should be applied at the boundary.
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100 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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101 % neighbour_boundary is a string specifying which boundary to interface to.
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102 function [closure, penalty] = boundary_condition(obj,boundary,type,data)
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103 default_arg('type','neumann');
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104 default_arg('data',0);
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105
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106 [e,d,s,gamm,halfnorm_inv] = obj.get_boundary_ops(boundary);
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107
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108 switch type
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109 % Dirichlet boundary condition
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110 case {'D','d','dirichlet'}
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111 alpha = obj.alpha;
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112
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113 % tau1 < -alpha^2/gamma
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114 tuning = 1.1;
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115 tau1 = -tuning*alpha/gamm;
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116 tau2 = s*alpha;
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117
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118 p = tau1*e + tau2*d;
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119
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120 closure = halfnorm_inv*p*e';
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121
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122 pp = halfnorm_inv*p;
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123 switch class(data)
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124 case 'double'
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125 penalty = pp*data;
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126 case 'function_handle'
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127 penalty = @(t)pp*data(t);
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128 otherwise
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129 error('Wierd data argument!')
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130 end
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131
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132
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133 % Neumann boundary condition
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134 case {'N','n','neumann'}
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135 alpha = obj.alpha;
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136 tau1 = -s*alpha;
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137 tau2 = 0;
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138 tau = tau1*e + tau2*d;
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139
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140 closure = halfnorm_inv*tau*d';
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141
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142 pp = halfnorm_inv*tau;
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143 switch class(data)
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144 case 'double'
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145 penalty = pp*data;
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146 case 'function_handle'
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147 penalty = @(t)pp*data(t);
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148 otherwise
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149 error('Wierd data argument!')
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150 end
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151
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152 % Unknown, boundary condition
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153 otherwise
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154 error('No such boundary condition: type = %s',type);
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155 end
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156 end
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157
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158 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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159 % u denotes the solution in the own domain
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160 % v denotes the solution in the neighbour domain
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161 [e_u,d_u,s_u,gamm_u, halfnorm_inv] = obj.get_boundary_ops(boundary);
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162 [e_v,d_v,s_v,gamm_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
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163
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164 tuning = 1.1;
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165
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166 alpha_u = obj.alpha;
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167 alpha_v = neighbour_scheme.alpha;
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168
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169 % tau1 < -(alpha_u/gamm_u + alpha_v/gamm_v)
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170
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171 tau1 = -(alpha_u/gamm_u + alpha_v/gamm_v) * tuning;
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172 tau2 = s_u*1/2*alpha_u;
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173 sig1 = s_u*(-1/2);
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174 sig2 = 0;
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175
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176 tau = tau1*e_u + tau2*d_u;
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177 sig = sig1*e_u + sig2*d_u;
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178
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179 closure = halfnorm_inv*( tau*e_u' + sig*alpha_u*d_u');
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180 penalty = halfnorm_inv*(-tau*e_v' - sig*alpha_v*d_v');
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181 end
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182
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183 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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184 % The right boundary is considered the positive boundary
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185 function [e,d,s,gamm, halfnorm_inv] = get_boundary_ops(obj,boundary)
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186 switch boundary
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187 case 'w'
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188 e = obj.e_w;
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189 d = obj.d1_w;
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190 s = -1;
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191 gamm = obj.gamm_x;
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192 halfnorm_inv = obj.Hix;
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193 case 'e'
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194 e = obj.e_e;
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195 d = obj.d1_e;
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196 s = 1;
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197 gamm = obj.gamm_x;
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198 halfnorm_inv = obj.Hix;
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199 case 's'
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200 e = obj.e_s;
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201 d = obj.d1_s;
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202 s = -1;
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203 gamm = obj.gamm_y;
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204 halfnorm_inv = obj.Hiy;
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205 case 'n'
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206 e = obj.e_n;
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207 d = obj.d1_n;
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208 s = 1;
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209 gamm = obj.gamm_y;
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210 halfnorm_inv = obj.Hiy;
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211 otherwise
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212 error('No such boundary: boundary = %s',boundary);
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213 end
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214 end
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215
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216 function N = size(obj)
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217 N = prod(obj.m);
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218 end
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219
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220 end
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221
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222 methods(Static)
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223 % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u
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224 % and bound_v of scheme schm_v.
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225 % [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l')
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226 function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v)
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227 [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v);
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228 [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u);
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229 end
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230 end
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231 end |
