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annotate +scheme/Euler1d.m @ 62:00344139cd7d

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Euler1d. Fixed mistake in last commit.
author Jonatan Werpers <jonatan@werpers.com>
date Sat, 14 Nov 2015 19:30:27 -0800
parents 183d9349b4c1
children 21b18edb9667
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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 M % Derivative norm
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12 gamma
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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, e_L, e_R;
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18 end
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19
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20 methods
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21 function obj = Euler1d(m,xlim,order,gama,opsGen,do_upwind)
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22 default_arg('opsGen',@sbp.Ordinary);
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23 default_arg('gama', 1.4);
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24 default_arg('do_upwind', false);
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25 gamma = gama;
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27 [x, h] = util.get_grid(xlim{:},m);
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29 if do_upwind
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30 ops = sbp.Upwind(m,h,order);
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31 Dp = ops.derivatives.Dp;
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32 Dm = ops.derivatives.Dm;
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34 D1 = (Dp + Dm)/2;
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35 Ddisp = (Dp - Dm)/2;
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36 else
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37 ops = opsGen(m,h,order);
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38 D1 = sparse(ops.derivatives.D1);
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39 end
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40
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41 H = sparse(ops.norms.H);
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42 Hi = sparse(ops.norms.HI);
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43 e_l = sparse(ops.boundary.e_1);
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44 e_r = sparse(ops.boundary.e_m);
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45
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46 I_x = speye(m);
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47 I_3 = speye(3);
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48
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49 D1 = kr(D1, I_3);
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50 if do_upwind
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51 Ddisp = kr(Ddisp,I_3);
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52 end
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53
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54 % Norms
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55 obj.H = kr(H,I_3);
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56 obj.Hi = kr(Hi,I_3);
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58 % Boundary operators
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59 obj.e_l = e_l;
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60 obj.e_r = e_r;
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61 obj.e_L = kr(e_l,I_3);
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62 obj.e_R = kr(e_r,I_3);
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64 obj.m = m;
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65 obj.h = h;
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66 obj.order = order;
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67
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68 % Man har Q_t+F_x=0 i 1D Euler, där
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69 % q=[rho, rho*u, e]^T
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70 % F=[rho*u, rho*u^2+p, (e+p)*u] ^T
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71 % p=(gamma-1)*(e-rho*u^2/2);
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73
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74 %Solving on form q_t + F_x = 0
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75
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76 function o = Fx(q)
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77 Q = reshape(q,3,m);
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78 o = reshape(obj.F(Q),3*m,1);
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79 o = D1*o;
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80 end
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81
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82 function o = Fx_disp(q)
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83 Q = reshape(q,3,m);
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84 f = reshape(obj.F(Q),3*m,1);
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85
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86 c = obj.c(Q);
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87 lambda_max = c+abs(Q(2,:)./Q(1,:));
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88 % lambda_max = max(lambda_max);
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89
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90 lamb_Q(1,:) = lambda_max.*Q(1,:);
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91 lamb_Q(2,:) = lambda_max.*Q(2,:);
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92 lamb_Q(3,:) = lambda_max.*Q(3,:);
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93
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94 lamb_q = reshape(lamb_Q,3*m, 1);
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95
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96 o = -D1*f + Ddisp*lamb_q;
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97 end
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98
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99 if do_upwind
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100 obj.D = @Fx_disp;
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101 else
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102 obj.D = @(q)-Fx(q);
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103 end
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104
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105 obj.u = x;
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106 obj.x = kr(x,ones(3,1));
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107 obj.gamma = gamma;
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108 end
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109
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110 % Flux function
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111 function o = F(obj, Q)
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112 % Flux: f = [q2; q2.^2/q1 + p(q); (q3+p(q))*q2/q1];
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113 p = obj.p(Q);
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114 o = [Q(2,:); Q(2,:).^2./Q(1,:) + p; (Q(3,:)+p).*Q(2,:)./Q(1,:)];
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115 end
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116
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117 % Equation of state
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118 function o = p(obj, Q)
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119 % Pressure p = (gamma-1)*(q3-q2.^2/q1/2)
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120 o = (obj.gamma-1)*( Q(3,:)-1/2*Q(2,:).^2./Q(1,:) );
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121 end
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122
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123 % Speed of sound
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124 function o = c(obj, Q)
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125 % Speed of light c = sqrt(obj.gamma*p/rho);
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126 o = sqrt(obj.gamma*obj.p(Q)./Q(1,:));
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127 end
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128
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129 % Eigen value matrix
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130 function o = Lambda(obj, q)
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131 u = q(2)/q(1);
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132 c = obj.c(q);
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133 L = [u, u+c, u-c];
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134 o = diag(L);
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135 end
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136
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137 % Diagonalization transformation
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138 function o = T(obj, q)
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139 % T is the transformation such that A = T*Lambda*inv(T)
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140 % where Lambda=diag(u, u+c, u-c)
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141 rho = q(1);
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142 u = q(2)/q(1);
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143 e = q(3);
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144 gamma = obj.gamma;
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145
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146 c = sqrt(gamma*obj.p(q)/rho);
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147
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148 sqrt2gamm = sqrt(2*(gamma-1));
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149
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150 o = [
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151 sqrt2gamm*rho , rho , rho ;
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152 sqrt2gamm*rho*u , rho*(u+c) , rho*(u-c) ;
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153 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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154 ];
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155 % Devide columns by stuff to get rid of extra comp?
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156 end
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157
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158 % Enforces the boundary conditions
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159 % w+ = R*w- + g(t)
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160 function closure = boundary_condition(obj,boundary, type, varargin)
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161 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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162
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163 % Boundary condition on form
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164 % w_in = R*w_out + g, where g is data
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165
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166 switch type
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167 case 'wall'
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168 closure = obj.boundary_condition_wall(boundary);
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169 case 'inflow'
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170 closure = obj.boundary_condition_inflow(boundary,varargin{:});
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171 case 'outflow'
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172 closure = obj.boundary_condition_outflow(boundary,varargin{:});
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173 case 'char'
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174 closure = obj.boundary_condition_char(boundary,varargin{:});
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175 otherwise
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176 error('Unsupported bc type: %s', type);
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177 end
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178
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179 end
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180
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181
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182 % Sets the boundary condition Lq = g, where
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183 % L = L(rho, u, e), g = g(t)
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184 % p_in are the indecies of the ingoing characteristics.
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185 function closure = boundary_condition_L(obj, boundary, L_fun, g_fun, p_in)
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186 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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187
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188 p_ot = 1:3;
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189 p_ot(p_in) = [];
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190
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191 p = [p_in, p_ot]; % Permutation to sort
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192 pt(p) = 1:length(p); % Inverse permutation
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193
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194 function o = closure_fun(q,t)
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195 % Extract solution at the boundary
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196 q_s = e_S'*q;
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197 rho = q_s(1);
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198 u = q_s(2)/rho;
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199 e = q_s(3);
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200
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201 c = obj.c(q_s);
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202
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203 % Calculate transformation matrix
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204 T = obj.T(q_s);
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205 Tin = T(:,p_in);
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206 Tot = T(:,p_ot);
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207
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208 % Convert bc from ordinary form to characteristic form.
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209 % Lq = g => w_in = Rw_ot + g_tilde
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210 L = L_fun(rho,u,e);
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211 g = g_fun(t);
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212
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213 Lambda = obj.Lambda(q_s);
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214
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215 R =-inv(L*Tin)*(L*Tot);
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216 g_tilde = inv(L*Tin)*g;
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217
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218 % Setup the penalty parameter
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219 tau1 = -2*Lambda(p_in,p_in);
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220 tau2 = zeros(length(p_ot),length(p_in)); % Penalty only on ingoing char.
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221
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222 tauHat = [tau1; tau2];
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223 tau = -s*e_S*sparse(T*tauHat(pt,:));
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224
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225 % Calculate the numerical caracteristics and apply the penalty
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226 w_s = inv(T)*q_s;
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227 % w_s = T\q_s;
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228 % w_s = Tinv * q_s; % Med analytisk matris
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229 w_in = w_s(p_in);
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230 w_ot = w_s(p_ot);
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231
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232 o = obj.Hi * tau * (w_in - R*w_ot - g_tilde);
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233 end
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234 closure = @closure_fun;
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235 end
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236
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237 function closure = boundary_condition_char(obj,boundary,w_data)
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238 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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239
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240 function o = closure_fun(q,t)
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241 q_s = e_S'*q;
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242 rho = q_s(1);
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243 u = q_s(2)/rho;
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244 e = q_s(3);
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245
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246 c = obj.c(q_s);
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247
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248 Lambda = [u, u+c, u-c];
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249
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250 p_in = find(s*Lambda < 0);
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251 p_ot = find(s*Lambda >= 0);
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252 p = [p_in p_ot];
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253 pt(p) = 1:length(p);
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254
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255 T = obj.T(q_s);
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256
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257 tau1 = -2*diag(Lambda(p_in));
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258 tau2 = zeros(length(p_ot),length(p_in)); % Penalty only on ingoing char.
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259
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260 tauHat = [tau1; tau2];
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261
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262 tau = -s*e_S*sparse(T*tauHat(pt,:));
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263
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264 w_s = inv(T)*q_s;
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265 w_in = w_s(p_in);
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266
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267 w_s_data = w_data(t);
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268 w_in_data = w_s_data(p_in);
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269
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270 o = obj.Hi * tau * (w_in - w_in_data);
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271 end
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272
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273 closure = @closure_fun;
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274
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275 end
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276
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diff changeset
277 function closure = boundary_condition_inflow(obj, boundary, p_data, v_data)
60
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278 [~,~,s] = obj.get_boundary_ops(boundary);
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279
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280 switch s
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281 case -1
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282 p_in = [1 2];
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283 case 1
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284 p_in = [1 3];
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285 end
49
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286
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287 a = obj.gamma - 1;
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288 L = @(~,u,~)[
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289 0 1 0;
e707cd9e6d0a Euler1d: Created a general bc routine.
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290 0 -1/2*u*a a;
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291 ];
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292 g = @(t)[
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diff changeset
293 v_data(t);
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294 p_data(t);
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295 ];
49
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296
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297 closure = boundary_condition_L(obj, boundary, L, g, p_in);
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298 end
0
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299
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300
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301 function closure = boundary_condition_outflow(obj, boundary, p_data)
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302 [~,~,s] = obj.get_boundary_ops(boundary);
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303
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304 switch s
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305 case -1
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306 p_in = 2;
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307 case 1
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308 p_in = 3;
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309 end
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310
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311 a = obj.gamma -1;
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312 L = @(~,u,~)a*[0 -1/2*u 1];
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313 g = @(t)[p_data(t)];
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314
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315
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316 closure = boundary_condition_L(obj, boundary, L, g, p_in);
0
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317
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318 end
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319
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320 % Set wall boundary condition v = 0.
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321 function closure = boundary_condition_wall(obj,boundary)
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322 [e_s,e_S,s] = obj.get_boundary_ops(boundary);
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323
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324 % v = 0 corresponds to
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325 % L = [0 1 0];
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326 % g = 0
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327 %
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328 % Tp =
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329 % R = -(u-c)/(u+c)
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330
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331 % tau = alpha * (u+c)
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332 % (alpha+1)(u+c) + 1/4* alpha^2|u-c| <= 0
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333 % 4*(alpha+1)(u+c) + alpha^2|u-c| <= 0
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334 % 4 * (alpha+1)(u+c) + alpha^2|u| + alpha^2*c <= 0
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335 % |u|*(sgn(u)*4 + sgn(u)*4*alpha + alpha^2) + c*(4 + 4alpha + alpha^2) <= 0
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336 % |u|*(alpha^2 + 4*sgn(u)*alpha + 4*sgn(u)) + c*(alpha+2)^2 <= 0
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337 % |u|*[(alpha + 2*sgn(u))^2 - 4*(sgn(u)-1)] + c*(alpha+2)^2 <= 0
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338
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339
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340 % om vi låtsas att u = 0:
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341 % (alpha+1)c + 1/4 * alpha^2*c <= 0
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342 % alpha^2 + 4*alpha +4 <= 0
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343 % (alpha + 2)^2 <= 0
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344 % alpha = -2 gives tau = -2*c;
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345
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346
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347 % Vill vi sätta penalty på karateristikan som är nära noll också eller vill
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348 % vi låta den vara fri?
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349
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350
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351 switch s
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352 case -1
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353 p_in = 2;
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354 p_zero = 1;
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355 p_ot = 3;
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356 case 1
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357 p_in = 3;
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358 p_zero = 1;
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359 p_ot = 2;
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360 otherwise
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361 error();
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362 end
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363
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364 p = [p_in, p_zero, p_ot]; % Permutation to sort
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365 pt(p) = 1:length(p); % Inverse permutation
8f0c2dc747dd Made lots of updates to Euler1d.
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366
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367 function o = closure_fun(q)
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368
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369 q_s = e_S'*q;
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370 rho = q_s(1);
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371 u = q_s(2)/rho;
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75ebf5d3cfe5 Performance improvement for Euler1d
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372 c = obj.c(q_s);
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373
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374 T = obj.T(q_s);
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375 R = -(u-c)/(u+c);
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376 % l = [u, u+c, u-c];
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377
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378 % p_in = find(s*l <= 0);
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379 % p_ot = find(s*l > 0);
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380
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381
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382 tau1 = -2*c;
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383 tau2 = [0; 0]; % Penalty only on ingoing char.
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384
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385 % Lambda_in = diag(abs(l(p_in)));
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386 % Lambda_ot = diag(abs(l(p_ot)));
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387
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388 tauHat = [tau1; tau2];
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389 tau = -s*e_S*sparse(T*tauHat(pt));
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390
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391 w_s = inv(T)*q_s;
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392 % w_s = T\q_s;
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393 % w_s = Tinv * q_s; % Med analytisk matris
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394 w_in = w_s(p_in);
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395 w_ot = w_s(p_ot);
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396
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397 o = obj.Hi * tau * (w_in - R*w_ot);
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398 end
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399
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400 closure = @closure_fun;
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401 end
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402
0
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403 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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404 error('NOT DONE')
0
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405 % u denotes the solution in the own domain
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406 % v denotes the solution in the neighbour domain
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407 [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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408 [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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409
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410 tuning = 2;
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411
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412 alpha_u = obj.alpha;
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413 alpha_v = neighbour_scheme.alpha;
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414
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415 tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning;
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416 % tau1 = (alpha_u/2 + alpha_v/2)/(2*delt_u)*tuning;
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417 tau4 = s_u*alpha_u/2;
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418
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419 sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning;
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420 sig3 = -s_u*alpha_u/2;
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421
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422 phi2 = s_u*1/2;
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423
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424 psi1 = -s_u*1/2;
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425
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426 tau = tau1*e_u + tau4*d3_u;
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427 sig = sig2*d1_u + sig3*d2_u ;
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428 phi = phi2*d1_u ;
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429 psi = psi1*e_u ;
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430
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431 closure = halfnorm_inv*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u');
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432 penalty = -halfnorm_inv*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v');
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433 end
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434
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435 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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436 % The right boundary is considered the positive boundary
49
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437 function [e,E,s] = get_boundary_ops(obj,boundary)
0
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438 switch boundary
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439 case 'l'
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440 e = obj.e_l;
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441 E = obj.e_L;
0
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442 s = -1;
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443 case 'r'
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444 e = obj.e_r;
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445 E = obj.e_R;
0
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446 s = 1;
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447 otherwise
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448 error('No such boundary: boundary = %s',boundary);
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449 end
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450 end
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451
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452 function N = size(obj)
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453 N = prod(obj.m);
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454 end
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455
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456 end
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457 end