solution.py 6.49 KB
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 """ Peter Creasey p.e.creasey.00@googlemail.com solution to the Sedov problem based on the C code by Aamer Haque """ from scipy.special import gamma as Gamma from numpy import power, arange, empty, float64, log, exp, pi, diff, inner, outer, array def calc_a(g,nu=3): """ exponents of the polynomials of the sedov solution g - the polytropic gamma nu - the dimension """ a = [0]*8 a[0] = 2.0 / (nu + 2) a[2] = (1-g) / (2*(g-1) + nu) a[3] = nu / (2*(g-1) + nu) a[5] = 2 / (g-2) a[6] = g / (2*(g-1) + nu) a[1] = (((nu+2)*g)/(2.0+nu*(g-1.0)) ) * ( (2.0*nu*(2.0-g))/(g*(nu+2.0)**2) - a[2]) a[4] = a[1]*(nu+2) / (2-g) a[7] = (2 + nu*(g-1))*a[1]/(nu*(2-g)) return a def calc_beta(v, g, nu=3): """ beta values for the sedov solution (coefficients of the polynomials of the similarity variables) v - the similarity variable g - the polytropic gamma nu- the dimension """ beta = (nu+2) * (g+1) * array((0.25, (g/(g-1))*0.5, -(2 + nu*(g-1))/2.0 / ((nu+2)*(g+1) -2*(2 + nu*(g-1))), -0.5/(g-1)), dtype=float64) beta = outer(beta, v) beta += (g+1) * array((0.0, -1.0/(g-1), (nu+2) / ((nu+2)*(g+1) -2.0*(2 + nu*(g-1))), 1.0/(g-1)), dtype=float64).reshape((4,1)) return beta def sedov(t, E0, rho0, g, n=1000, nu=3): """ solve the sedov problem t - the time E0 - the initial energy rho0 - the initial density n - number of points (10000) nu - the dimension g - the polytropic gas gamma """ # the similarity variable v_min = 2.0 / ((nu + 2) * g) v_max = 4.0 / ((nu + 2) * (g + 1)) v = v_min + arange(n) * (v_max - v_min) / (n - 1.0) a = calc_a(g, nu) beta = calc_beta(v, g=g, nu=nu) lbeta = log(beta) r = exp(-a[0] * lbeta[0] - a[2] * lbeta[1] - a[1] * lbeta[2]) rho = ((g + 1.0) / (g - 1.0)) * exp(a[3] * lbeta[1] + a[5] * lbeta[3] + a[4] * lbeta[2]) p = exp(nu * a[0] * lbeta[0] + (a[5] + 1) * lbeta[3] + (a[4] - 2 * a[1]) * lbeta[2]) u = beta[0] * r * 4.0 / ((g + 1) * (nu + 2)) p *= 8.0 / ((g + 1) * (nu + 2) * (nu + 2)) # we have to take extra care at v=v_min, since this can be a special point. # It is not a singularity, however, the gradients of our variables (wrt v) are. # r -> 0, u -> 0, rho -> 0, p-> constant u[0] = 0.0; rho[0] = 0.0; r[0] = 0.0; p[0] = p[1] # volume of an n-sphere vol = (pi ** (nu / 2.0) / Gamma(nu / 2.0 + 1)) * power(r, nu) # note we choose to evaluate the integral in this way because the # volumes of the first few elements (i.e near v=vmin) are shrinking # very slowly, so we dramatically improve the error convergence by # finding the volumes exactly. This is most important for the # pressure integral, as this is on the order of the volume. # (dimensionless) energy of the model solution de = rho * u * u * 0.5 + p / (g - 1) # integrate (trapezium rule) q = inner(de[1:] + de[:-1], diff(vol)) * 0.5 # the factor to convert to this particular problem fac = (q * (t ** nu) * rho0 / E0) ** (-1.0 / (nu + 2)) # shock speed shock_speed = fac * (2.0 / (nu + 2)) rho_s = ((g + 1) / (g - 1)) * rho0 r_s = shock_speed * t * (nu + 2) / 2.0 p_s = (2.0 * rho0 * shock_speed * shock_speed) / (g + 1) u_s = (2.0 * shock_speed) / (g + 1) r *= fac * t u *= fac p *= fac * fac * rho0 rho *= rho0 return r, p, rho, u, r_s, p_s, rho_s, u_s, shock_speed def test(): """ draw a 3d sedov solution """ import pylab as pl gamma = 5.0/3.0 r,p,rho,u,r_s,p_s,rho_s,u_s,shock_speed = \ sedov(t=0.05, E0=5.0, rho0=5.0, g=gamma) print 'rho shock', rho_s print 'p shock', p_s print 'u shock', u_s print 'r shock', r_s print 'Dimensionless var (E/rho) t^2 r^-5', (5.0 /5.0)* 0.05**0.4 * r[-1]**-1.0 vols = (4/3.0)*pi*r*r*r dv = vols.copy() dv[1:] = diff(dv) # thermal and kinetic energy te = (p*dv/(gamma-1)) ke = (rho*u*u*0.5*dv) energy = te.sum() + ke.sum() mass = 0.5*inner(rho[1:]+rho[:-1],dv[1:]) print 'density', mass / (4/3.0 * pi * r_s**3) print 'energy', energy print 'shock speed', shock_speed pl.plot(r/r_s,rho/rho_s, label=r'$\rho/\rho_s$') pl.plot(r/r_s,p/p_s, label=r'$p/p_s$') pl.plot(r/r_s,u/u_s, label=r'$u/u_s$') pl.legend(loc='upper left') pl.show() def test2(): """ test momentum and mass conservation in 3d """ import pylab as pl r,p,rho,u,r_s,p_s,rho_s,u_s,shock_speed = \ sedov(t=0.05, E0=5.0, rho0=5.0, g=5.0/3.0,n=10000) dt = 1e-5 r2,p2,rho2,u2 = sedov(t=0.05+dt, E0=5.0, rho0=5.0, g=5.0/3.0, n=9000)[:4] # align the results from numpy import interp, gradient p2 = interp(r,r2,p2) rho2 = interp(r,r2,rho2) u2 = interp(r,r2,u2) # mass conservation pl.plot(r, -gradient(rho*u*r*r)/(r*r*gradient(r)), 'b', label=r'$\frac{1}{r^2}\frac{\partial}{\partial r} \rho u r^2$') pl.plot(r, (rho2-rho)/dt, 'k', label=r'$\frac{\partial \rho}{\partial t}$') # momentum conservation pl.plot(r, -gradient(p)/gradient(r), 'g',label=r'$-\frac{\partial p}{\partial r}$') pl.plot(r, rho*((u2-u)/dt+u*gradient(u)/gradient(r)), 'r',label=r'$\rho \left( \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial r} \right)$') pl.legend(loc='lower left') pl.show() def test3(): """ draw a 2d sedov solution """ import pylab as pl r,p,rho,u,r_s,p_s,rho_s,u_s,shock_speed = sedov(t=1.2, E0=1, rho0=1, g=5.0/3.0, nu=2) print 'rho shock', rho_s print 'p shock', p_s print 'u shock', u_s print 'r shock', r_s area = pi*r*r dv = area.copy() dv[1:] = diff(dv) # thermal and kinetic energy te = (p*dv/(5.0/3.0-1)) ke = (rho*u*u*0.5*dv) #pl.plot(arange(te.size), ke, 'x') #pl.show() print 'r0', r[:2] energy = te.sum() + ke.sum() mass = 0.5*inner(rho[1:]+rho[:-1],dv[1:]) print 'density', mass / (pi * r_s**2) print 'energy', energy print 'shock speed', shock_speed #pl.plot(r/r_s,rho/rho_s, 'b,',label=r'$\rho/\rho_s$') #pl.plot(r/r_s,p/p_s,'r',label=r'$p/p_s$') #pl.plot(r/r_s,u/u_s, 'g,',label=r'$u/u_s$') # pl.plot(r,rho,'b',label='$\\rho$') pl.plot(r,u,'b',label='$u$') # pl.plot(r,p,'b',label='$P$') pl.legend(loc='upper left') # pl.xlim(0,2) # pl.ylim(0,5) pl.show() if __name__=='__main__':  Pedro Gonnet committed Mar 19, 2013 212  test()  213  #test2()  Pedro Gonnet committed Mar 19, 2013 214  # test3()