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1 classdef SchmBeam2d < 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 N % Number of points total
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5 h % Grid spacing
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6 u,v % Grid
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7 x,y % Values of x and y for each grid point
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8 order % Order accuracy for the approximation
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9
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10 D % non-stabalized scheme operator
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11 M % Derivative norm
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12 alpha
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13
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14 H % Discrete norm
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15 Hi
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16 H_x, H_y % Norms in the x and y directions
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17 Hx,Hy % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
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18 Hi_x, Hi_y
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19 Hix, Hiy
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20 e_w, e_e, e_s, e_n
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21 d1_w, d1_e, d1_s, d1_n
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22 d2_w, d2_e, d2_s, d2_n
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23 d3_w, d3_e, d3_s, d3_n
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24 gamm_x, gamm_y
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25 delt_x, delt_y
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26 end
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27
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28 methods
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29 function obj = SchmBeam2d(m,lim,order,alpha,opsGen)
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30 default_arg('opsGen',@sbp.Higher);
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31 default_arg('a',1);
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32
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33 if length(m) == 1
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34 m = [m m];
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35 end
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36
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37 m_x = m(1);
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38 m_y = m(2);
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39
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40 xlim = lim{1};
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41 ylim = lim{2};
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42
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43 [x, h_x] = util.get_grid(xlim{:},m_x);
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44 [y, h_y] = util.get_grid(ylim{:},m_y);
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45
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46 ops_x = opsGen(m_x,h_x,order);
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47 ops_y = opsGen(m_y,h_y,order);
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48
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49 I_x = speye(m_x);
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50 I_y = speye(m_y);
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51
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52
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53
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54
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55 D4_x = sparse(ops_x.derivatives.D4);
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56 H_x = sparse(ops_x.norms.H);
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57 Hi_x = sparse(ops_x.norms.HI);
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58 e_l_x = sparse(ops_x.boundary.e_1);
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59 e_r_x = sparse(ops_x.boundary.e_m);
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60 d1_l_x = sparse(ops_x.boundary.S_1);
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61 d1_r_x = sparse(ops_x.boundary.S_m);
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62 d2_l_x = sparse(ops_x.boundary.S2_1);
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63 d2_r_x = sparse(ops_x.boundary.S2_m);
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64 d3_l_x = sparse(ops_x.boundary.S3_1);
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65 d3_r_x = sparse(ops_x.boundary.S3_m);
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66
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67 D4_y = sparse(ops_y.derivatives.D4);
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68 H_y = sparse(ops_y.norms.H);
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69 Hi_y = sparse(ops_y.norms.HI);
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70 e_l_y = sparse(ops_y.boundary.e_1);
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71 e_r_y = sparse(ops_y.boundary.e_m);
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72 d1_l_y = sparse(ops_y.boundary.S_1);
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73 d1_r_y = sparse(ops_y.boundary.S_m);
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74 d2_l_y = sparse(ops_y.boundary.S2_1);
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75 d2_r_y = sparse(ops_y.boundary.S2_m);
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76 d3_l_y = sparse(ops_y.boundary.S3_1);
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77 d3_r_y = sparse(ops_y.boundary.S3_m);
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78
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79
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80 D4 = kr(D4_x, I_y) + kr(I_x, D4_y);
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81
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82 % Norms
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83 obj.H = kr(H_x,H_y);
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84 obj.Hx = kr(H_x,I_x);
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85 obj.Hy = kr(I_x,H_y);
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86 obj.Hix = kr(Hi_x,I_y);
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87 obj.Hiy = kr(I_x,Hi_y);
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88 obj.Hi = kr(Hi_x,Hi_y);
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89
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90 % Boundary operators
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91 obj.e_w = kr(e_l_x,I_y);
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92 obj.e_e = kr(e_r_x,I_y);
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93 obj.e_s = kr(I_x,e_l_y);
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94 obj.e_n = kr(I_x,e_r_y);
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95 obj.d1_w = kr(d1_l_x,I_y);
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96 obj.d1_e = kr(d1_r_x,I_y);
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97 obj.d1_s = kr(I_x,d1_l_y);
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98 obj.d1_n = kr(I_x,d1_r_y);
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99 obj.d2_w = kr(d2_l_x,I_y);
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100 obj.d2_e = kr(d2_r_x,I_y);
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101 obj.d2_s = kr(I_x,d2_l_y);
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102 obj.d2_n = kr(I_x,d2_r_y);
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103 obj.d3_w = kr(d3_l_x,I_y);
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104 obj.d3_e = kr(d3_r_x,I_y);
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105 obj.d3_s = kr(I_x,d3_l_y);
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106 obj.d3_n = kr(I_x,d3_r_y);
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107
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108 obj.m = m;
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109 obj.h = [h_x h_y];
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110 obj.order = order;
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111
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112 obj.alpha = alpha;
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113 obj.D = alpha*D4;
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114 obj.u = x;
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115 obj.v = y;
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116 obj.x = kr(x,ones(m_y,1));
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117 obj.y = kr(ones(m_x,1),y);
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118
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119 obj.gamm_x = h_x*ops_x.borrowing.N.S2/2;
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120 obj.delt_x = h_x^3*ops_x.borrowing.N.S3/2;
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121
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122 obj.gamm_y = h_y*ops_y.borrowing.N.S2/2;
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123 obj.delt_y = h_y^3*ops_y.borrowing.N.S3/2;
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124 end
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125
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126
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127 % Closure functions return the opertors applied to the own doamin to close the boundary
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128 % 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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129 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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130 % type is a string specifying the type of boundary condition if there are several.
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131 % data is a function returning the data that should be applied at the boundary.
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132 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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133 % neighbour_boundary is a string specifying which boundary to interface to.
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134 function [closure, penalty_e,penalty_d] = boundary_condition(obj,boundary,type,data)
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135 default_arg('type','dn');
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136 default_arg('data',0);
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137
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138 [e,d1,d2,d3,s,gamm,delt,halfnorm_inv] = obj.get_boundary_ops(boundary);
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139
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140 switch type
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141 % Dirichlet-neumann boundary condition
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142 case {'dn'}
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143 alpha = obj.alpha;
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144
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145 % tau1 < -alpha^2/gamma
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146 tuning = 1.1;
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147
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148 tau1 = tuning * alpha/delt;
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149 tau4 = s*alpha;
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150
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151 sig2 = tuning * alpha/gamm;
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152 sig3 = -s*alpha;
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153
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154 tau = tau1*e+tau4*d3;
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155 sig = sig2*d1+sig3*d2;
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156
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157 closure = halfnorm_inv*(tau*e' + sig*d1');
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158
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159 pp_e = halfnorm_inv*tau;
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160 pp_d = halfnorm_inv*sig;
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161 switch class(data)
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162 case 'double'
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163 penalty_e = pp_e*data;
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164 penalty_d = pp_d*data;
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165 case 'function_handle'
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166 penalty_e = @(t)pp_e*data(t);
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167 penalty_d = @(t)pp_d*data(t);
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168 otherwise
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169 error('Wierd data argument!')
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170 end
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171
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172 % Unknown, boundary condition
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173 otherwise
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174 error('No such boundary condition: type = %s',type);
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175 end
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176 end
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177
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178 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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179 % u denotes the solution in the own domain
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180 % v denotes the solution in the neighbour domain
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181 [e_u,d1_u,d2_u,d3_u,s_u,gamm_u,delt_u, halfnorm_inv] = obj.get_boundary_ops(boundary);
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182 [e_v,d1_v,d2_v,d3_v,s_v,gamm_v,delt_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
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183
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184 tuning = 2;
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185
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186 alpha_u = obj.alpha;
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187 alpha_v = neighbour_scheme.alpha;
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188
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189 tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning;
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190 % tau1 = (alpha_u/2 + alpha_v/2)/(2*delt_u)*tuning;
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191 tau4 = s_u*alpha_u/2;
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192
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193 sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning;
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194 sig3 = -s_u*alpha_u/2;
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195
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196 phi2 = s_u*1/2;
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197
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198 psi1 = -s_u*1/2;
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199
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200 tau = tau1*e_u + tau4*d3_u;
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201 sig = sig2*d1_u + sig3*d2_u ;
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202 phi = phi2*d1_u ;
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203 psi = psi1*e_u ;
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204
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205 closure = halfnorm_inv*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u');
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206 penalty = -halfnorm_inv*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v');
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207 end
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208
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209 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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210 % The right boundary is considered the positive boundary
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211 function [e,d1,d2,d3,s,gamm, delt, halfnorm_inv] = get_boundary_ops(obj,boundary)
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212 switch boundary
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213 case 'w'
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214 e = obj.e_w;
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215 d1 = obj.d1_w;
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216 d2 = obj.d2_w;
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217 d3 = obj.d3_w;
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218 s = -1;
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219 gamm = obj.gamm_x;
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220 delt = obj.delt_x;
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221 halfnorm_inv = obj.Hix;
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222 case 'e'
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223 e = obj.e_e;
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224 d1 = obj.d1_e;
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225 d2 = obj.d2_e;
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226 d3 = obj.d3_e;
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227 s = 1;
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228 gamm = obj.gamm_x;
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229 delt = obj.delt_x;
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230 halfnorm_inv = obj.Hix;
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231 case 's'
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232 e = obj.e_s;
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233 d1 = obj.d1_s;
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234 d2 = obj.d2_s;
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235 d3 = obj.d3_s;
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236 s = -1;
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237 gamm = obj.gamm_y;
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238 delt = obj.delt_y;
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239 halfnorm_inv = obj.Hiy;
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240 case 'n'
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241 e = obj.e_n;
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242 d1 = obj.d1_n;
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243 d2 = obj.d2_n;
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244 d3 = obj.d3_n;
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245 s = 1;
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246 gamm = obj.gamm_y;
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247 delt = obj.delt_y;
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248 halfnorm_inv = obj.Hiy;
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249 otherwise
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250 error('No such boundary: boundary = %s',boundary);
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251 end
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252 end
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253
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254 function N = size(obj)
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255 N = prod(obj.m);
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256 end
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257
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258 end
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259 end
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