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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 % Grid values
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7 x % Values of x and y for each
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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 e_l, e_r
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17
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18 end
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19
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20 methods
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21 function obj = SchmBeam2d(m,xlim,order,gamma,opsGen)
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22 default_arg('opsGen',@sbp.Ordinary);
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23 default_arg('gamma', 1.4);
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24
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25 [x, h] = util.get_grid(xlim{:},m_x);
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26
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27 ops = opsGen(m_x,h_x,order);
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28
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29 I_x = speye(m);
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30 I_3 = speye(3);
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31
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32 D1 = sparse(ops.derivatives.D1);
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33 H = sparse(ops.norms.H);
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34 Hi = sparse(ops.norms.HI);
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35 e_l = sparse(ops.boundary.e_1);
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36 e_r = sparse(ops.boundary.e_m);
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37
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38 D1 = kr(D1, I_3);
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39
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40 % Norms
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41 obj.H = kr(H,I_3);
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42
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43 % Boundary operators
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44 obj.e_l = kr(e_l,I_3);
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45 obj.e_r = kr(e_r,I_3);
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46
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47 obj.m = m;
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48 obj.h = h;
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49 obj.order = order;
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50
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51
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52 % Man har Q_t+F_x=0 i 1D Euler, där
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53 % q=[rho, rho*u, e]^T
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54 % F=[rho*u, rho*u^2+p, (e+p)*u] ^T
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55 % p=(gamma-1)*(e-rho/2*u^2);
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56
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57
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58 %Solving on form q_t + F_x = 0
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59 function o = F(q)
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60 o = [q(2); q(2).^2/q(1) + p(q); (q(3)+p(q))*q(2)/q(1)];
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61 end
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62
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63 % Equation of state
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64 function o = p(q)
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65 o = (gamma-1)*(q(3)-q(2).^2/q(1)/2);
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66 end
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67
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68
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69 % R =
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70 % [sqrt(2*(gamma-1))*rho , rho , rho ;
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71 % sqrt(2*(gamma-1))*rho*u , rho*(u+c) , rho*(u-c) ;
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72 % sqrt(2*(gamma-1))*rho*u^2/2, e+(gamma-1)*(e-rho*u^2/2)+rho*u*c, e+(gamma-1)*(e-rho*u^2/2)-rho*u*c]);
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73 function o = R(q)
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74 rho = q(1);
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75 u = q(2)/q(1);
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76 e = q(3);
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77
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78 sqrt2gamm = sqrt(2*(gamma-1));
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79
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80 o = [
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81 sqrt2gamm*rho , rho , rho ;
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82 sqrt2gamm*rho*u , rho*(u+c) , rho*(u-c) ;
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83 sqrt2gamm*rho*u^2/2, e+(gamma-1)*(e-rho*u^2/2)+rho*u*c , e+(gamma-1)*(e-rho*u^2/2)-rho*u*c
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84 ];
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85 end
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86
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87 function o = Fx(q)
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88 o = zeros(size(q));
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89 for i = 1:3:3*m
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90 o(i:i+2) = F(q(i:i+2));
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91 end
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92 end
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93
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94
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95
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96 % A=R*Lambda*inv(R), där Lambda=diag(u, u+c, u-c) (c är ljudhastigheten)
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97 % c^2=gamma*p/rho
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98 % function o = A(rho,u,e)
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99 % end
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100
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101
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102 obj.D = @Fx;
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103 obj.u = x;
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104 obj.x = kr(x,ones(3,1));
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105 end
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106
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107
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108 % Closure functions return the opertors applied to the own doamin to close the boundary
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109 % 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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110 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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111 % type is a string specifying the type of boundary condition if there are several.
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112 % data is a function returning the data that should be applied at the boundary.
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113 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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114 % neighbour_boundary is a string specifying which boundary to interface to.
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115 function [closure, penalty] = boundary_condition(obj,boundary, alpha,data)
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116 default_arg('alpha',0);
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117 default_arg('data',0);
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118
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119 % Boundary condition on form
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120 % w_in = w_out + g, where g is data
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121
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122 [e,s] = obj.get_boundary_ops(boundary);
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123
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124 tuning = 1; % ?????????????????????????
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125
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126 tau = R(q)*lambda(q)*tuning; % SHOULD THIS BE abs(lambda)?????
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127
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128 function closure_fun(q,t)
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129 q_b = e * q;
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130 end
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131
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132 function penalty_fun(q,t)
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133 end
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134
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135
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136
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137
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138
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139 % tau1 < -alpha^2/gamma
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140
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141 tau1 = tuning * alpha/delt;
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142 tau4 = s*alpha;
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143
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144 sig2 = tuning * alpha/gamm;
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145 sig3 = -s*alpha;
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146
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147 tau = tau1*e+tau4*d3;
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148 sig = sig2*d1+sig3*d2;
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149
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150 closure = halfnorm_inv*(tau*e' + sig*d1');
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151
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152 pp_e = halfnorm_inv*tau;
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153 pp_d = halfnorm_inv*sig;
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154 switch class(data)
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155 case 'double'
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156 penalty_e = pp_e*data;
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157 penalty_d = pp_d*data;
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158 case 'function_handle'
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159 penalty_e = @(t)pp_e*data(t);
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160 penalty_d = @(t)pp_d*data(t);
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161 otherwise
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162 error('Wierd data argument!')
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163 end
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164
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165 end
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166
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167 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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168 % u denotes the solution in the own domain
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169 % v denotes the solution in the neighbour domain
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170 [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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171 [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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172
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173 tuning = 2;
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174
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175 alpha_u = obj.alpha;
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176 alpha_v = neighbour_scheme.alpha;
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177
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178 tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning;
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179 % tau1 = (alpha_u/2 + alpha_v/2)/(2*delt_u)*tuning;
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180 tau4 = s_u*alpha_u/2;
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181
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182 sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning;
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183 sig3 = -s_u*alpha_u/2;
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184
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185 phi2 = s_u*1/2;
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186
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187 psi1 = -s_u*1/2;
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188
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189 tau = tau1*e_u + tau4*d3_u;
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190 sig = sig2*d1_u + sig3*d2_u ;
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191 phi = phi2*d1_u ;
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192 psi = psi1*e_u ;
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193
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194 closure = halfnorm_inv*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u');
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195 penalty = -halfnorm_inv*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v');
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196 end
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197
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198 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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199 % The right boundary is considered the positive boundary
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200 function [e,d1,d2,d3,s,gamm, delt, halfnorm_inv] = get_boundary_ops(obj,boundary)
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201 switch boundary
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202 case 'w'
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203 e = obj.e_w;
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204 d1 = obj.d1_w;
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205 d2 = obj.d2_w;
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206 d3 = obj.d3_w;
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207 s = -1;
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208 gamm = obj.gamm_x;
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209 delt = obj.delt_x;
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210 halfnorm_inv = obj.Hix;
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211 case 'e'
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212 e = obj.e_e;
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213 d1 = obj.d1_e;
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214 d2 = obj.d2_e;
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215 d3 = obj.d3_e;
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216 s = 1;
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217 gamm = obj.gamm_x;
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218 delt = obj.delt_x;
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219 halfnorm_inv = obj.Hix;
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220 case 's'
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221 e = obj.e_s;
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222 d1 = obj.d1_s;
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223 d2 = obj.d2_s;
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224 d3 = obj.d3_s;
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225 s = -1;
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226 gamm = obj.gamm_y;
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227 delt = obj.delt_y;
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228 halfnorm_inv = obj.Hiy;
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229 case 'n'
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230 e = obj.e_n;
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231 d1 = obj.d1_n;
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232 d2 = obj.d2_n;
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233 d3 = obj.d3_n;
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234 s = 1;
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235 gamm = obj.gamm_y;
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236 delt = obj.delt_y;
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237 halfnorm_inv = obj.Hiy;
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238 otherwise
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239 error('No such boundary: boundary = %s',boundary);
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240 end
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241 end
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242
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243 function N = size(obj)
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244 N = prod(obj.m);
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245 end
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246
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247 end
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248 end
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