Added the 2D Sedov test-case

examples/GreshoVortex/plotSolution.py

@@ -93,7 +93,7 @@ git = sim["Code"].attrs["Git Revision"]
 
 pos = sim["/PartType0/Coordinates"][:,:]
 x = pos[:,0] - boxSize / 2
-y = pos[:, 1] - boxSize / 2
+y = pos[:,1] - boxSize / 2
 vel = sim["/PartType0/Velocities"][:,:]
 r = sqrt(x**2 + y**2)
 v_r = (x * vel[:,0] + y * vel[:,1]) / r

examples/GreshoVortex/run.sh

 #!/bin/bash
 
  # Generate the initial conditions if they are not present.
-if [ ! -e glass_128.hdf5 ]
+if [ ! -e glassPlane_128.hdf5 ]
 then
     echo "Fetching initial glass file for the Gresho-Chan vortex example..."
     ./getGlass.sh

examples/SedovBlast_2D/getGlass.sh

+#!/bin/bash
+wget http://virgodb.cosma.dur.ac.uk/swift-webstorage/ICs/glassPlane_128.hdf5

examples/SedovBlast_2D/makeIC.py

+###############################################################################
+ # This file is part of SWIFT.
+ # Copyright (c) 2016 Matthieu Schaller (matthieu.schaller@durham.ac.uk)
+ # 
+ # This program is free software: you can redistribute it and/or modify
+ # it under the terms of the GNU Lesser General Public License as published
+ # by the Free Software Foundation, either version 3 of the License, or
+ # (at your option) any later version.
+ # 
+ # This program is distributed in the hope that it will be useful,
+ # but WITHOUT ANY WARRANTY; without even the implied warranty of
+ # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ # GNU General Public License for more details.
+ # 
+ # You should have received a copy of the GNU Lesser General Public License
+ # along with this program.  If not, see <http://www.gnu.org/licenses/>.
+ # 
+ ##############################################################################
+
+import h5py
+import random
+from numpy import *
+
+# Generates a swift IC file for the Sedov blast test in a periodic cubic box
+
+# Parameters
+gamma = 5./3.      # Gas adiabatic index
+rho0 = 1.          # Background density
+P0 = 1.e-6         # Background pressure
+E0= 1.             # Energy of the explosion
+N_inject = 21      # Number of particles in which to inject energy
+fileName = "sedov.hdf5" 
+
+#L = 101
+
+#---------------------------------------------------
+glass = h5py.File("glassPlane_128.hdf5", "r")
+
+# Read particle positions and h from the glass
+pos = glass["/PartType0/Coordinates"][:,:]
+h = glass["/PartType0/SmoothingLength"][:] * 0.3
+
+numPart = size(h)
+vol = 1.
+
+# Generate extra arrays
+v = zeros((numPart, 3))
+ids = linspace(1, numPart, numPart)
+m = zeros(numPart)
+u = zeros(numPart)
+r = zeros(numPart)
+
+for i in range(numPart):
+        r[i] = sqrt((pos[i,0] - 0.5)**2 + (pos[i,1] - 0.5)**2)
+        m[i] = rho0 * vol / numPart    
+        u[i] = P0 / (rho0 * (gamma - 1))
+
+# Make the central particle detonate
+index = argsort(r)
+u[index[0:N_inject]] = E0 / (N_inject * m[0])
+
+#--------------------------------------------------
+
+#File
+file = h5py.File(fileName, 'w')
+
+# Header
+grp = file.create_group("/Header")
+grp.attrs["BoxSize"] = [1., 1., 1.0]
+grp.attrs["NumPart_Total"] =  [numPart, 0, 0, 0, 0, 0]
+grp.attrs["NumPart_Total_HighWord"] = [0, 0, 0, 0, 0, 0]
+grp.attrs["NumPart_ThisFile"] = [numPart, 0, 0, 0, 0, 0]
+grp.attrs["Time"] = 0.0
+grp.attrs["NumFilesPerSnapshot"] = 1
+grp.attrs["MassTable"] = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+grp.attrs["Flag_Entropy_ICs"] = 0
+
+#Runtime parameters
+grp = file.create_group("/RuntimePars")
+grp.attrs["PeriodicBoundariesOn"] = 1
+
+#Units
+grp = file.create_group("/Units")
+grp.attrs["Unit length in cgs (U_L)"] = 1.
+grp.attrs["Unit mass in cgs (U_M)"] = 1.
+grp.attrs["Unit time in cgs (U_t)"] = 1.
+grp.attrs["Unit current in cgs (U_I)"] = 1.
+grp.attrs["Unit temperature in cgs (U_T)"] = 1.
+
+#Particle group
+grp = file.create_group("/PartType0")
+grp.create_dataset('Coordinates', data=pos, dtype='d')
+grp.create_dataset('Velocities', data=v, dtype='f')
+grp.create_dataset('Masses', data=m, dtype='f')
+grp.create_dataset('SmoothingLength', data=h, dtype='f')
+grp.create_dataset('InternalEnergy', data=u, dtype='f')
+grp.create_dataset('ParticleIDs', data=ids, dtype='L')
+
+file.close()

examples/SedovBlast_2D/plotSolution.py

+###############################################################################
+ # This file is part of SWIFT.
+ # Copyright (c) 2015 Bert Vandenbroucke (bert.vandenbroucke@ugent.be)
+ #                    Matthieu Schaller (matthieu.schaller@durham.ac.uk)
+ # 
+ # This program is free software: you can redistribute it and/or modify
+ # it under the terms of the GNU Lesser General Public License as published
+ # by the Free Software Foundation, either version 3 of the License, or
+ # (at your option) any later version.
+ # 
+ # This program is distributed in the hope that it will be useful,
+ # but WITHOUT ANY WARRANTY; without even the implied warranty of
+ # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ # GNU General Public License for more details.
+ # 
+ # You should have received a copy of the GNU Lesser General Public License
+ # along with this program.  If not, see <http://www.gnu.org/licenses/>.
+ # 
+ ##############################################################################
+
+# Computes the analytical solution of the 3D Sedov blast wave.
+# The script works for a given initial box and dumped energy and computes the solution at a later time t.
+
+# Parameters
+rho_0 = 1.          # Background Density
+P_0 = 1.e-6         # Background Pressure
+E_0 = 1.            # Energy of the explosion
+gas_gamma = 5./3.   # Gas polytropic index
+
+
+# ---------------------------------------------------------------
+# Don't touch anything after this.
+# ---------------------------------------------------------------
+
+import matplotlib
+matplotlib.use("Agg")
+from pylab import *
+import h5py
+
+# Plot parameters
+params = {'axes.labelsize': 10,
+'axes.titlesize': 10,
+'font.size': 12,
+'legend.fontsize': 12,
+'xtick.labelsize': 10,
+'ytick.labelsize': 10,
+'text.usetex': True,
+ 'figure.figsize' : (9.90,6.45),
+'figure.subplot.left'    : 0.045,
+'figure.subplot.right'   : 0.99,
+'figure.subplot.bottom'  : 0.05,
+'figure.subplot.top'     : 0.99,
+'figure.subplot.wspace'  : 0.15,
+'figure.subplot.hspace'  : 0.12,
+'lines.markersize' : 6,
+'lines.linewidth' : 3.,
+'text.latex.unicode': True
+}
+rcParams.update(params)
+rc('font',**{'family':'sans-serif','sans-serif':['Times']})
+
+
+snap = int(sys.argv[1])
+
+
+# Read the simulation data
+sim = h5py.File("sedov_%03d.hdf5"%snap, "r")
+boxSize = sim["/Header"].attrs["BoxSize"][0]
+time = sim["/Header"].attrs["Time"][0]
+scheme = sim["/HydroScheme"].attrs["Scheme"]
+kernel = sim["/HydroScheme"].attrs["Kernel function"]
+neighbours = sim["/HydroScheme"].attrs["Kernel target N_ngb"]
+eta = sim["/HydroScheme"].attrs["Kernel eta"]
+git = sim["Code"].attrs["Git Revision"]
+
+pos = sim["/PartType0/Coordinates"][:,:]
+x = pos[:,0] - boxSize / 2
+y = pos[:,1] - boxSize / 2
+vel = sim["/PartType0/Velocities"][:,:]
+r = sqrt(x**2 + y**2)
+v_r = (x * vel[:,0] + y * vel[:,1]) / r
+u = sim["/PartType0/InternalEnergy"][:]
+S = sim["/PartType0/Entropy"][:]
+P = sim["/PartType0/Pressure"][:]
+rho = sim["/PartType0/Density"][:]
+
+
+from scipy.special import gamma as Gamma
+from numpy import *
+
+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
+
+r_s, P_s, rho_s, v_s, r_shock, _, _, _, _ = sedov(time, E_0, rho_0, gas_gamma, 1000, 2)
+
+# Append points for after the shock
+r_s = np.insert(r_s, np.size(r_s), [r_shock, r_shock*1.5])
+rho_s = np.insert(rho_s, np.size(rho_s), [rho_0, rho_0])
+P_s = np.insert(P_s, np.size(P_s), [P_0, P_0])
+v_s = np.insert(v_s, np.size(v_s), [0, 0])
+
+# Additional arrays
+u_s = P_s / (rho_s * (gas_gamma - 1.))  #internal energy
+s_s = P_s / rho_s**gas_gamma # entropic function
+
+# Plot the interesting quantities
+figure()
+
+# Velocity profile --------------------------------
+subplot(231)
+plot(r, v_r, '.', color='r', ms=1.)
+plot(r_s, v_s, '--', color='k', alpha=0.8, lw=1.2)
+xlabel("${\\rm{Radius}}~r$", labelpad=0)
+ylabel("${\\rm{Radial~velocity}}~v_r$", labelpad=0)
+xlim(0, 1.3 * r_shock)
+ylim(-0.2, 3.8)
+
+# Density profile --------------------------------
+subplot(232)
+plot(r, rho, '.', color='r', ms=1.)
+plot(r_s, rho_s, '--', color='k', alpha=0.8, lw=1.2)
+xlabel("${\\rm{Radius}}~r$", labelpad=0)
+ylabel("${\\rm{Density}}~\\rho$", labelpad=2)
+xlim(0, 1.3 * r_shock)
+ylim(-0.2, 5.2)
+
+# Pressure profile --------------------------------
+subplot(233)
+plot(r, P, '.', color='r', ms=1.)
+plot(r_s, P_s, '--', color='k', alpha=0.8, lw=1.2)
+xlabel("${\\rm{Radius}}~r$", labelpad=0)
+ylabel("${\\rm{Pressure}}~P$", labelpad=0)
+xlim(0, 1.3 * r_shock)
+ylim(-1, 12.5)
+
+# Internal energy profile -------------------------
+subplot(234)
+plot(r, u, '.', color='r', ms=1.)
+plot(r_s, u_s, '--', color='k', alpha=0.8, lw=1.2)
+xlabel("${\\rm{Radius}}~r$", labelpad=0)
+ylabel("${\\rm{Internal~Energy}}~u$", labelpad=0)
+xlim(0, 1.3 * r_shock)
+ylim(0, 120)
+
+# Entropy profile ---------------------------------
+subplot(235)
+plot(r, S, '.', color='r', ms=1.)
+plot(r_s, s_s, '--', color='k', alpha=0.8, lw=1.2)
+xlabel("${\\rm{Radius}}~r$", labelpad=0)
+ylabel("${\\rm{Entropy}}~S$", labelpad=-5)
+xlim(0, 1.3 * r_shock)
+ylim(0, 1000)
+
+# Information -------------------------------------
+subplot(236, frameon=False)
+
+text(-0.49, 0.9, "Sedov blast with  $\\gamma=%.3f$ in 2D at $t=%.2f$"%(gas_gamma,time), fontsize=10)
+text(-0.49, 0.8, "Background $\\rho_0=%.2f$"%(rho_0), fontsize=10)
+text(-0.49, 0.7, "Energy injected $E_0=%.2f$"%(E_0), fontsize=10)
+plot([-0.49, 0.1], [0.62, 0.62], 'k-', lw=1)
+text(-0.49, 0.5, "$\\textsc{Swift}$ %s"%git, fontsize=10)
+text(-0.49, 0.4, scheme, fontsize=10)
+text(-0.49, 0.3, kernel, fontsize=10)
+text(-0.49, 0.2, "$%.2f$ neighbours ($\\eta=%.3f$)"%(neighbours, eta), fontsize=10)
+xlim(-0.5, 0.5)
+ylim(0, 1)
+xticks([])
+yticks([])
+
+
+savefig("Sedov.png", dpi=200)
+
+
+
+

examples/SedovBlast_2D/run.sh

+#!/bin/bash
+
+ # Generate the initial conditions if they are not present.
+if [ ! -e glassPlane_128.hdf5 ]
+then
+    echo "Fetching initial glass file for the Sedov blast example..."
+    ./getGlass.sh
+fi
+if [ ! -e sedov.hdf5 ]
+then
+    echo "Generating initial conditions for the Sedov blast example..."
+    python makeIC.py
+fi
+
+# Run SWIFT
+../swift -s -t 1 sedov.yml
+
+# Plot the solution
+python plotSolution.py 5

examples/SedovBlast_2D/sedov.py

+###############################################################################
+ # This file is part of SWIFT.
+ # Copyright (c) 2015 Bert Vandenbroucke (bert.vandenbroucke@ugent.be)
+ # 
+ # This program is free software: you can redistribute it and/or modify
+ # it under the terms of the GNU Lesser General Public License as published
+ # by the Free Software Foundation, either version 3 of the License, or
+ # (at your option) any later version.
+ # 
+ # This program is distributed in the hope that it will be useful,
+ # but WITHOUT ANY WARRANTY; without even the implied warranty of
+ # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ # GNU General Public License for more details.
+ # 
+ # You should have received a copy of the GNU Lesser General Public License
+ # along with this program.  If not, see <http://www.gnu.org/licenses/>.
+ # 
+ ##############################################################################
+
+import numpy as np
+import scipy.integrate as integrate
+import scipy.optimize as optimize
+import os
+
+# Calculate the analytical solution of the Sedov-Taylor shock wave for a given
+# number of dimensions, gamma and time. We assume dimensionless units and a
+# setup with constant density 1, pressure and velocity 0. An energy 1 is
+# inserted in the center at t 0.
+#
+# The solution is a self-similar shock wave, which was described in detail by
+# Sedov (1959). We follow his notations and solution method.
+#
+# The position of the shock at time t is given by
+#  r2 = (E/rho1)^(1/(2+nu)) * t^(2/(2+nu))
+# the energy E is related to the inserted energy E0 by E0 = alpha*E, with alpha
+# a constant which has to be calculated by imposing energy conservation.
+#
+# The density for a given radius at a certain time is determined by introducing
+# the dimensionless position coordinate lambda = r/r2. The density profile as
+# a function of lambda is constant in time and given by
+#  rho = rho1 * R(V, gamma, nu)
+# and
+#  V = V(lambda, gamma, nu)
+#
+# The function V(lambda, gamma, nu) is found by solving a differential equation
+# described in detail in Sedov (1959) chapter 4, section 5. Alpha is calculated
+# from the integrals in section 11 of the same chapter.
+#
+# Numerically, the complete solution requires the use of 3 quadratures and 1
+# root solver, which are implemented using the GNU Scientific Library (GSL).
+# Since some quadratures call functions that themselves contain quadratures,
+# the problem is highly non-linear and complex and takes a while to converge.
+# Therefore, we tabulate the alpha values and profile values the first time
+# a given set of gamma and nu values is requested and reuse these tabulated
+# values.
+#
+# Reference:
+#  Sedov (1959): Sedov, L., Similitude et dimensions en mecanique (7th ed.;
+#                Moscou: Editions Mir) - french translation of the original
+#                book from 1959.
+
+# dimensionless variable z = gamma*P/R as a function of V (and gamma and nu)
+# R is a dimensionless density, while P is a dimensionless pressure
+# z is hence a sort of dimensionless sound speed
+# The expression below corresponds to eq. 11.9 in Sedov (1959), chapter 4
+def _z(V, gamma, nu):
+    if V == 2./(nu+2.)/gamma:
+        return 0.
+    else:
+        return (gamma-1.)*V*V*(V-2./(2.+nu))/2./(2./(2.+nu)/gamma-V)
+
+# differential equation that needs to be solved to obtain lambda for a given V
+# corresponds to eq. 5.11 in Sedov (1959), chapter 4 (omega = 0)
+def _dlnlambda_dV(V, gamma, nu):
+    nom = _z(V, gamma, nu) - (V-2./(nu+2.))*(V-2./(nu+2.))
+    denom = V*(V-1.)*(V-2./(nu+2.))+nu*(2./(nu+2.)/gamma-V)*_z(V, gamma, nu)
+    return nom/denom
+
+# dimensionless variable lambda = r/r2 as a function of V (and gamma and nu)
+# found by solving differential equation 5.11 in Sedov (1959), chapter 4
+# (omega = 0)
+def _lambda(V, gamma, nu):
+    if V == 2./(nu+2.)/gamma:
+        return 0.
+    else:
+        V0 = 4./(nu+2.)/(gamma+1.)
+        integral, err = integrate.quad(_dlnlambda_dV, V, V0, (gamma, nu),
+                                       limit = 8000)
+        return np.exp(-integral)
+
+# dimensionless variable R = rho/rho1 as a function of V (and gamma and nu)
+# found by inverting eq. 5.12 in Sedov (1959), chapter 4 (omega = 0)
+# the integration constant C is found by inserting the R, V and z values
+# at the shock wave, where lambda is 1. These correspond to eq. 11.8 in Sedov
+# (1959), chapter 4.
+def _R(V, gamma, nu):
+    if V == 2./(nu+2.)/gamma:
+        return 0.
+    else:
+        C = 8.*gamma*(gamma-1.)/(nu+2.)/(nu+2.)/(gamma+1.)/(gamma+1.) \
+                *((gamma-1.)/(gamma+1.))**(gamma-2.) \
+                *(4./(nu+2.)/(gamma+1.)-2./(nu+2.))
+        lambda1 = _lambda(V, gamma, nu)
+        lambda5 = lambda1**(nu+2)
+        return (_z(V, gamma, nu)*(V-2./(nu+2.))*lambda5/C)**(1./(gamma-2.))
+
+# function of which we need to find the zero point to invert lambda(V)
+def _lambda_min_lambda(V, lambdax, gamma, nu):
+    return _lambda(V, gamma, nu) - lambdax
+
+# dimensionless variable V = v*t/r as a function of lambda (and gamma and nu)
+# found by inverting the function lamdba(V) which is found by solving
+# differential equation 5.11 in Sedov (1959), chapter 4 (omega = 0)
+# the invertion is done by searching the zero point of the function
+# lambda_min_lambda defined above
+def _V_inv(lambdax, gamma, nu):
+    if lambdax == 0.:
+        return 2./(2.+nu)/gamma;
+    else:
+        return optimize.brentq(_lambda_min_lambda, 2./(nu+2.)/gamma, 
+                               4./(nu+2.)/(gamma+1.), (lambdax, gamma, nu))
+
+# integrand of the first integral in eq. 11.24 in Sedov (1959), chapter 4
+def _integrandum1(lambdax, gamma, nu):
+    V = _V_inv(lambdax, gamma, nu)
+    if nu == 2:
+        return _R(V, gamma, nu)*V**2*lambdax**3
+    else:
+        return _R(V, gamma, nu)*V**2*lambdax**4
+
+# integrand of the second integral in eq. 11.24 in Sedov (1959), chapter 4
+def _integrandum2(lambdax, gamma, nu):
+    V = _V_inv(lambdax, gamma, nu)
+    if V == 2./(nu+2.)/gamma:
+        P = 0.
+    else:
+        P = _z(V, gamma, nu)*_R(V, gamma, nu)/gamma
+    if nu == 2:
+        return P*lambdax**3
+    else:
+        return P*lambdax**4
+
+# calculate alpha = E0/E
+# this corresponds to eq. 11.24 in Sedov (1959), chapter 4
+def get_alpha(gamma, nu):
+  integral1, err1 = integrate.quad(_integrandum1, 0., 1., (gamma, nu))
+  integral2, err2 = integrate.quad(_integrandum2, 0., 1., (gamma, nu))
+  
+  if nu == 2:
+    return np.pi*integral1+2.*np.pi/(gamma-1.)*integral2
+  else:
+    return 2.*np.pi*integral1+4.*np.pi/(gamma-1.)*integral2
+
+# get the analytical solution for the Sedov-Taylor blastwave given an input
+# energy E, adiabatic index gamma, and number of dimensions nu, at time t and
+# with a maximal outer region radius maxr
+def get_analytical_solution(E, gamma, nu, t, maxr = 1.):
+    # we check for the existance of a datafile with precomputed alpha and
+    # profile values
+    # if it does not exist, we calculate it here and write it out
+    # calculation of alpha and the profile takes a while...
+    lvec = np.zeros(1000)
+    Rvec = np.zeros(1000)
+    fname = "sedov_profile_gamma_{gamma}_nu_{nu}.dat".format(gamma = gamma,
+                                                             nu = nu)
+    if os.path.exists(fname):
+        file = open(fname, "r")
+        lines = file.readlines()
+        alpha = float(lines[0])
+        for i in range(len(lines)-1):
+            data = lines[i+1].split()
+            lvec[i] = float(data[0])
+            Rvec[i] = float(data[1])
+    else:
+        alpha = get_alpha(gamma, nu)
+        for i in range(1000):
+            lvec[i] = (i+1)*0.001
+            V = _V_inv(lvec[i], gamma, nu)
+            Rvec[i] = _R(V, gamma, nu)
+        file = open(fname, "w")
+        file.write("#{alpha}\n".format(alpha = alpha))
+        for i in range(1000):
+            file.write("{l}\t{R}\n".format(l = lvec[i], R = Rvec[i]))
+
+    xvec = np.zeros(1002)
+    rhovec = np.zeros(1002)
+    if nu == 2:
+        r2 = (E/alpha)**0.25*np.sqrt(t)
+    else:
+        r2 = (E/alpha)**0.2*t**0.4
+
+    for i in range(1000):
+        xvec[i] = lvec[i]*r2
+        rhovec[i] = Rvec[i]
+    xvec[1000] = 1.001*r2
+    xvec[1001] = maxr
+    rhovec[1000] = 1.
+    rhovec[1001] = 1.
+    
+    return xvec, rhovec
+
+def main():
+    E = 1.
+    gamma = 1.66667
+    nu = 2
+    t = 0.001
+    x, rho = get_analytical_solution(E, gamma, nu, t)
+    for i in range(len(x)):
+        print x[i], rho[i]
+
+if __name__ == "__main__":
+    main()

examples/SedovBlast_2D/sedov.yml

+# Define the system of units to use internally. 
+InternalUnitSystem:
+  UnitMass_in_cgs:     1   # Grams
+  UnitLength_in_cgs:   1   # Centimeters
+  UnitVelocity_in_cgs: 1   # Centimeters per second
+  UnitCurrent_in_cgs:  1   # Amperes
+  UnitTemp_in_cgs:     1   # Kelvin
+
+# Parameters governing the time integration
+TimeIntegration:
+  time_begin: 0.    # The starting time of the simulation (in internal units).
+  time_end:   5e-2  # The end time of the simulation (in internal units).
+  dt_min:     1e-7  # The minimal time-step size of the simulation (in internal units).
+  dt_max:     1e-4  # The maximal time-step size of the simulation (in internal units).
+
+# Parameters governing the snapshots
+Snapshots:
+  basename:            sedov # Common part of the name of output files
+  time_first:          0.    # Time of the first output (in internal units)
+  delta_time:          1e-2  # Time difference between consecutive outputs (in internal units)
+
+# Parameters governing the conserved quantities statistics
+Statistics:
+  delta_time:          1e-3 # Time between statistics output
+
+# Parameters for the hydrodynamics scheme
+SPH:
+  resolution_eta:        1.2348   # Target smoothing length in units of the mean inter-particle separation (1.2348 == 48Ngbs with the cubic spline kernel).
+  delta_neighbours:      0.1      # The tolerance for the targetted number of neighbours.
+  max_smoothing_length:  0.1       # Maximal smoothing length allowed (in internal units).
+  CFL_condition:         0.1      # Courant-Friedrich-Levy condition for time integration.
+
+# Parameters related to the initial conditions
+InitialConditions:
+  file_name:  ./sedov.hdf5          # The file to read
+