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annotate +scheme/Euler1d.m @ 54:2194cd385419

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Euler1d: Fixed russian dissipation. Cleaned up. Added non-working BC.
author Jonatan Werpers <jonatan@werpers.com>
date Wed, 11 Nov 2015 19:21:38 -0800
parents 75ebf5d3cfe5
children 9a647dcccbdd
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1 classdef Euler1d < scheme.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 Fx
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12 M % Derivative norm
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13 gamma
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14
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15 T
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16 p
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17 c
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20 H % Discrete norm
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21 Hi
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22 e_l, e_r, e_L, e_R;
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23
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24 end
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25
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26 methods
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27 function obj = Euler1d(m,xlim,order,gama,opsGen,do_upwind)
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28 default_arg('opsGen',@sbp.Ordinary);
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29 default_arg('gama', 1.4);
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30 default_arg('do_upwind', false);
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31 gamma = gama;
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32
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33 [x, h] = util.get_grid(xlim{:},m);
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35 if do_upwind
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36 ops = sbp.Upwind(m,h,order);
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37 Dp = ops.derivatives.Dp;
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38 Dm = ops.derivatives.Dm;
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40 D1 = (Dp + Dm)/2;
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41 Ddisp = (Dp - Dm)/2;
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42 else
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43 ops = opsGen(m,h,order);
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44 D1 = sparse(ops.derivatives.D1);
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45 end
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46
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47 H = sparse(ops.norms.H);
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48 Hi = sparse(ops.norms.HI);
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49 e_l = sparse(ops.boundary.e_1);
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50 e_r = sparse(ops.boundary.e_m);
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51
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52 I_x = speye(m);
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53 I_3 = speye(3);
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54
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55 D1 = kr(D1, I_3);
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56 if do_upwind
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57 Ddisp = kr(Ddisp,I_3);
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58 end
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59
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60 % Norms
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61 obj.H = kr(H,I_3);
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62 obj.Hi = kr(Hi,I_3);
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63
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64 % Boundary operators
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65 obj.e_l = e_l;
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66 obj.e_r = e_r;
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67 obj.e_L = kr(e_l,I_3);
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68 obj.e_R = kr(e_r,I_3);
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69
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70 obj.m = m;
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71 obj.h = h;
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72 obj.order = order;
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73
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74 % Man har Q_t+F_x=0 i 1D Euler, där
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75 % q=[rho, rho*u, e]^T
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76 % F=[rho*u, rho*u^2+p, (e+p)*u] ^T
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77 % p=(gamma-1)*(e-rho*u^2/2);
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78
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79 %Solving on form q_t + F_x = 0
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80 function o = F(Q)
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81 % Flux: f = [q2; q2.^2/q1 + p(q); (q3+p(q))*q2/q1];
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82 o = [Q(2,:); Q(2,:).^2/Q(1,:) + p(Q); (Q(3,:)+p(Q)).*Q(2,:)./Q(1,:)];
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83 end
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84
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85 % Equation of state
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86 function o = p(Q)
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87 % Pressure p = (gamma-1)*(q3-q2.^2/q1/2)
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88 o = (gamma-1)*(Q(3,:)-Q(2,:).^2/Q(1,:)/2);
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89 end
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90
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91 function o = c(Q)
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92 % Speed of light c = sqrt(obj.gamma*p/rho);
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93 o = sqrt(gamma*p(Q)./Q(1,:));
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94 end
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95
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96 function o = T(q)
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97 % T is the transformation such that A = T*Lambda*inv(T)
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98 % where Lambda=diag(u, u+c, u-c)
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99 rho = q(1);
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100 u = q(2)/q(1);
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101 e = q(3);
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102
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103 c = sqrt(gamma*p(q)/rho);
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104
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105 sqrt2gamm = sqrt(2*(gamma-1));
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106
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107 o = [
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108 sqrt2gamm*rho , rho , rho ;
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109 sqrt2gamm*rho*u , rho*(u+c) , rho*(u-c) ;
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110 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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111 ];
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112 % Devide columns by stuff to get rid of extra comp?
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113 end
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114
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115 function o = Fx(q)
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116 Q = reshape(q,3,m);
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117 o = reshape(F(Q),3*m,1);
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118 o = D1*o;
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119 end
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120
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121 function o = Fx_disp(q)
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122 Q = reshape(q,3,m);
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123 f = reshape(F(Q),3*m,1);
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124
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125 c = c(Q);
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126 lambda_max = c+abs(Q(2,:)./Q(1,:));
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127 % lambda_max = max(lambda_max);
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128
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129 lamb_Q(1,:) = lambda_max.*Q(1,:);
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130 lamb_Q(2,:) = lambda_max.*Q(2,:);
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131 lamb_Q(3,:) = lambda_max.*Q(3,:);
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132
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133 lamb_q = reshape(lamb_Q,3*m, 1);
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134
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135 o = -D1*f + Ddisp*lamb_q;
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136 end
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137
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138 if do_upwind
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139 obj.D = @Fx_disp;
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140 else
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141 obj.D = @(q)-Fx(q);
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142 end
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143
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144 obj.Fx = @Fx;
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145 obj.T = @T;
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146 obj.u = x;
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147 obj.x = kr(x,ones(3,1));
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148 obj.p = @p;
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149 obj.c = @c;
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150 obj.gamma = gamma;
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151 end
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152
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153
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154 % Enforces the boundary conditions
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155 % w+ = R*w- + g(t)
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156 function closure = boundary_condition(obj,boundary, type, varargin)
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157 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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158
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159 % Boundary condition on form
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160 % w_in = R*w_out + g, where g is data
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161
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162 switch type
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163 case 'wall'
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164 closure = obj.boundary_condition_wall(boundary);
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165 case 'inflow'
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166 closure = obj.boundary_condition_inflow(boundary,varargin{:});
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167 case 'outflow'
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168 closure = obj.boundary_condition_outflow(boundary,varargin{:});
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169 otherwise
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170 error('Unsupported bc type: %s', type);
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171 end
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172
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173 end
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174
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175 function closure = boundary_condition_inflow(obj, boundary, p_data, v_data)
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176 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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177
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178 switch s
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179 case -1
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180 p_in = [1 2];
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181 p_ot = [3];
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182 case 1
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183 p_in = [1 3];
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184 p_ot = [2];
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185 otherwise
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186 error();
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187 end
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188
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189 p = [p_in, p_ot]; % Permutation to sort
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190 pt(p) = 1:length(p); % Inverse permutation
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191
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192 gamma = obj.gamma;
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193
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194 function o = closure_fun(q,t)
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195 q_s = e_S'*q;
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196 rho = q_s(1);
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197 u = q_s(2)/rho;
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198 e = q_s(3);
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199
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200 c = obj.c(q_s);
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201
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202 T = obj.T(q_s);
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203
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204 % L = [
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205 % 0 1 0;
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206 % 0 -1/2*u 1;
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207 % ];
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208 LTpi = [
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209 2^(1/2)/(2*rho*u*(gamma - 1)^(1/2)), -(2^(1/2)*(u - c*s))/(u*(2*e*gamma - rho*(gamma*u^2 - c*s*u + 2*c*s))*(gamma - 1)^(1/2));
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210 0, 2/(2*e*gamma - gamma*rho*u^2 + c*rho*s*(u - 2));
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211 ];
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212 R = [
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213 (2^(1/2)*c*s*(rho*(gamma - 1)*u^2 + 2*rho*u - 2*e*gamma))/(u*(gamma - 1)^(1/2)*(- gamma*rho*u^2 + c*rho*s*u + 2*e*gamma - 2*c*rho*s));
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214 (2*c*rho*s*(u - 2))/(2*e*gamma - gamma*rho*u^2 + c*rho*s*(u - 2)) - 1;
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215 ];
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216
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217 tau1 = -2*[
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218 u, 0;
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219 0, c + abs(u);
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220 ];
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221 tau2 = [0 0]; % Penalty only on ingoing char.
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222
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223
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224 tauHat = [tau1; tau2];
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225 tau = -s*e_S*sparse(T*tauHat(pt,:));
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226
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227 w_s = inv(T)*q_s;
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228 % w_s = T\q_s;
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229 % w_s = Tinv * q_s; % Med analytisk matris
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230 w_in = w_s(p_in);
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231 w_ot = w_s(p_ot);
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232
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233
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234 o = obj.Hi * tau * (w_in - R*w_ot - LTpi*[v_data(t); p_data(t)]);
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235 end
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236
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237 closure = @closure_fun;
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238 end
0
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239
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240
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241 function closure = boundary_condition_outflow(obj, boundary, p_data)
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242 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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243
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244 switch s
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245 case -1
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246 p_in = 2;
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247 p_ot = [1 3];
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248 case 1
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249 p_in = 3;
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250 p_ot = [1 2];
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251 otherwise
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252 error();
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253 end
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254
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255 p = [p_in, p_ot]; % Permutation to sort
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256 pt(p) = 1:length(p); % Inverse permutation
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257
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258 gamma = obj.gamma;
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259
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260 function o = closure_fun(q,t)
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261 q_s = e_S'*q;
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262 rho = q_s(1);
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263 u = q_s(2)/rho;
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264 e = q_s(3);
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265
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266 c = obj.c(q_s);
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267
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268
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269 T = obj.T(q_s);
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270
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271 L = (gamma -1)*[0 -1/2*u 1];
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272 LTpi = 1/(L*T(:,p_in));
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273 R = -[0 (gamma*(e - rho*u^2/2) - rho*u*c)/(gamma*(e - rho*u^2/2) + rho*u*c)];
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274
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275 tau1 = [-2*(c - abs(u))];
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276 tau2 = [0; 0]; % Penalty only on ingoing char.
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277
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278
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279 tauHat = [tau1; tau2];
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280 tau = -s*e_S*sparse(T*tauHat(pt));
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281
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282 w_s = inv(T)*q_s;
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283 % w_s = T\q_s;
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284 % w_s = Tinv * q_s; % Med analytisk matris
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285 w_in = w_s(p_in);
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286 w_ot = w_s(p_ot);
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287
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288 o = obj.Hi * tau * (w_in - R*w_ot - LTpi*p_data(t));
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289 end
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290
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291 closure = @closure_fun;
0
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292
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293 end
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294
49
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295 % Set wall boundary condition v = 0.
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296 function closure = boundary_condition_wall(obj,boundary)
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297 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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298
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299 % v = 0 corresponds to
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300 % L = [0 1 0];
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301 % g = 0
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302 %
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303 % Tp =
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304 % R = -(u-c)/(u+c)
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305
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306 % tau = alpha * (u+c)
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307 % (alpha+1)(u+c) + 1/4* alpha^2|u-c| <= 0
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308 % 4*(alpha+1)(u+c) + alpha^2|u-c| <= 0
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309 % 4 * (alpha+1)(u+c) + alpha^2|u| + alpha^2*c <= 0
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310 % |u|*(sgn(u)*4 + sgn(u)*4*alpha + alpha^2) + c*(4 + 4alpha + alpha^2) <= 0
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311 % |u|*(alpha^2 + 4*sgn(u)*alpha + 4*sgn(u)) + c*(alpha+2)^2 <= 0
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312 % |u|*[(alpha + 2*sgn(u))^2 - 4*(sgn(u)-1)] + c*(alpha+2)^2 <= 0
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313
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314
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315 % om vi låtsas att u = 0:
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316 % (alpha+1)c + 1/4 * alpha^2*c <= 0
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317 % alpha^2 + 4*alpha +4 <= 0
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318 % (alpha + 2)^2 <= 0
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319 % alpha = -2 gives tau = -2*c;
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320
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321
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322 % Vill vi sätta penalty på karateristikan som är nära noll också eller vill
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323 % vi låta den vara fri?
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324
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325
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326 switch s
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327 case -1
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328 p_in = 2;
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329 p_zero = 1;
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330 p_ot = 3;
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331 case 1
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332 p_in = 3;
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333 p_zero = 1;
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334 p_ot = 2;
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335 otherwise
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336 error();
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337 end
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338
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339 p = [p_in, p_zero, p_ot]; % Permutation to sort
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340 pt(p) = 1:length(p); % Inverse permutation
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341
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342 function o = closure_fun(q)
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343
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344 q_s = e_S'*q;
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345 rho = q_s(1);
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346 u = q_s(2)/rho;
50
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347 c = obj.c(q_s);
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348
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349 T = obj.T(q_s);
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350 R = -(u-c)/(u+c);
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351 % l = [u, u+c, u-c];
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352
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353 % p_in = find(s*l <= 0);
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354 % p_ot = find(s*l > 0);
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355
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356
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357 tau1 = -2*c;
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358 tau2 = [0; 0]; % Penalty only on ingoing char.
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359
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360 % Lambda_in = diag(abs(l(p_in)));
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361 % Lambda_ot = diag(abs(l(p_ot)));
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362
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363 tauHat = [tau1; tau2];
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364 tau = -s*e_S*sparse(T*tauHat(pt));
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365
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366 w_s = inv(T)*q_s;
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367 % w_s = T\q_s;
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368 % w_s = Tinv * q_s; % Med analytisk matris
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369 w_in = w_s(p_in);
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370 w_ot = w_s(p_ot);
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371
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372 o = obj.Hi * tau * (w_in - R*w_ot);
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373 end
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374
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375 closure = @closure_fun;
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376 end
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377
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378 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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379 error('NOT DONE')
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380 % u denotes the solution in the own domain
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381 % v denotes the solution in the neighbour domain
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382 [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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383 [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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384
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385 tuning = 2;
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386
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387 alpha_u = obj.alpha;
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388 alpha_v = neighbour_scheme.alpha;
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389
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390 tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning;
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391 % tau1 = (alpha_u/2 + alpha_v/2)/(2*delt_u)*tuning;
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392 tau4 = s_u*alpha_u/2;
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393
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394 sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning;
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395 sig3 = -s_u*alpha_u/2;
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396
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397 phi2 = s_u*1/2;
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398
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399 psi1 = -s_u*1/2;
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400
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401 tau = tau1*e_u + tau4*d3_u;
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402 sig = sig2*d1_u + sig3*d2_u ;
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403 phi = phi2*d1_u ;
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404 psi = psi1*e_u ;
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405
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406 closure = halfnorm_inv*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u');
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407 penalty = -halfnorm_inv*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v');
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408 end
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409
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410 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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411 % The right boundary is considered the positive boundary
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412 function [e,E,s] = get_boundary_ops(obj,boundary)
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413 switch boundary
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414 case 'l'
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415 e = obj.e_l;
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416 E = obj.e_L;
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417 s = -1;
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418 case 'r'
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419 e = obj.e_r;
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420 E = obj.e_R;
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421 s = 1;
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422 otherwise
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423 error('No such boundary: boundary = %s',boundary);
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424 end
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425 end
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Jonatan Werpers <jonatan@werpers.com>
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426
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Jonatan Werpers <jonatan@werpers.com>
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427 function N = size(obj)
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428 N = prod(obj.m);
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429 end
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430
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431 end
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432 end