def test_RadHydroMMS(self):
# declare symbolic variables
x, t, alpha, c, a, mu = symbols('x t alpha c a mu')
# number of refinement cycles
n_cycles = 5
# number of elements in first cycle
n_elems = 20
# We will increase sigma and nondimensional C as we decrease the mesh
# size to ensure we stay in the diffusion limit
# numeric values
gamma_value = 5. / 3.
cv_value = 0.14472799784454
# run for a fixed amount of time
t_end = 2. * pi
sig_s = 0.0
#Want material speeed to be a small fraction of speed of light
#and radiation to be small relative to kinetic energy
P = 0.001 # a_r T_inf^4/(rho_inf*u_inf^2)
#Arbitrary ratio of pressure to density
alpha_value = 0.5
#Arbitrary mach number well below the sound speed. The choice of gamma and
#the cv value, as well as C and P constrain all other material reference
#parameters, but we are free to choose the material velocity below the sound
#speed to ensure no shocks are formed
M = 0.9
cfl_value = 0.6 #but 2 time steps, so really like a CFL of 0.3
dt = []
dx = []
err = []
for cycle in range(n_cycles):
#Keep ratio of and sig_a*dx constant, staying in diffusion limit
C = sig_a = 100000. / 20 * n_elems # c/a_inf
sig_t = sig_s + sig_a
a_inf = GC.SPD_OF_LGT / C
#These lines of code assume specified C_v
T_inf = a_inf**2 / (gamma_value * (gamma_value - 1.) * cv_value)
rho_inf = GC.RAD_CONSTANT * T_inf**4 / (P * a_inf**2)
#Specify rho_inf and determine T_inf and Cv_value
rho_inf = 1.0 #If we don't choose a reference density, then the specific heat values get ridiculous
T_inf = pow(rho_inf * P * a_inf**2 / GC.RAD_CONSTANT,
0.25) #to set T_inf based on rho_inf,
cv_value = a_inf**2 / (T_inf * gamma_value * (gamma_value - 1.)
) # to set c_v, if rho specified
#Specify T_inf and solve for rho_inf and C_v value
# T_inf = 1
# rho_inf = GC.RAD_CONSTANT*T_inf**4/(P*a_inf**2)
# cv_value = a_inf**2/(T_inf*gamma_value*(gamma_value-1.)) # to set c_v, if rho specified
#Same for all approachs
p_inf = rho_inf * a_inf * a_inf
print "The dimensionalization parameters are: "
print " a_inf : ", a_inf
print " T_inf : ", T_inf
print " rho_inf : ", rho_inf
print " p_inf : ", p_inf
print " C_v : ", cv_value
print " gamma : ", gamma_value
# create solution for thermodynamic state and flow field
rho = rho_inf * (2. + sin(x - t))
u = a_inf * M * 0.5 * (2. + cos(x - t))
p = alpha_value * p_inf * (2. + cos(x - t))
e = p / (rho * (gamma_value - 1.))
E = 0.5 * rho * u * u + rho * e
# create solution for radiation field based on solution for F
# that is the leading order diffusion limit solution
T = e / cv_value
Er = a * T**4
Fr = -1. / (3. * sig_t) * c * diff(Er,
x) + sympify('4./3.') * Er * u
#Form psi+ and psi- from Fr and Er
psip = (Er * c * mu + Fr) / (2. * mu)
psim = (Er * c * mu - Fr) / (2. * mu)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
substitutions['a'] = GC.RAD_CONSTANT
substitutions['mu'] = RU.mu["+"]
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho * u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
T = T.subs(substitutions)
rho_f = lambdify((symbols('x'), symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'), symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'), symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'), symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'), symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'), symbols('t')), psip, "numpy")
T_f = lambdify((symbols('x'), symbols('t')), T, "numpy")
Er = Er.subs(substitutions)
Er = lambdify((symbols('x'), symbols('t')), Er, "numpy")
#For reference create lambda as ratio of aT^4/rho--u^2 to check
#P_f = lambda x,t: GC.RAD_CONSTANT*T_f(x,t)**4/(rho_f(x,t)*u_f(x,t)**2)
#dx = 1./100.
#x = -1.*dx
#t=0.2
#for i in range(100):
# x += dx
# print "ratio:", P_f(x,t)
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho=rho,
u=u,
E=E,
psim=psim,
psip=psip,
sigma_s_value=sig_s,
sigma_a_value=sig_a,
gamma_value=gamma_value,
cv_value=cv_value,
alpha_value=alpha_value,
display_equations=True)
# mesh
width = 2. * pi
mesh = Mesh(n_elems, width)
# compute hydro IC for the sake of computing initial time step size
hydro_IC = computeAnalyticHydroSolution(mesh,
t=0.0,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# compute the initial time step size according to CFL conditon (actually
# half). We will then decrease this time step by the same factor as DX each
# cycle
if cycle == 0:
sound_speed = [
sqrt(i.p * i.gamma / i.rho) + abs(i.u) for i in hydro_IC
]
dt_vals = [
cfl_value * (mesh.elements[i].dx) / sound_speed[i]
for i in xrange(len(hydro_IC))
]
dt_value = min(min(dt_vals),
0.5) #don't take too big of time step
print "initial dt_value", dt_value
#Adjust the end time to be an exact increment of dt_values
print "old t_end: ", t_end
print "new t_end: ", t_end
print "This cycle's dt value: ", dt_value
# create uniform mesh
mesh = Mesh(n_elems, width)
dt.append(dt_value)
dx.append(mesh.getElement(0).dx)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'periodic')
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,
t=0.0,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# Dimensionless parameters. These are all evaluated at peaks of trig
# functions, very hard coded
print "---------------------------------------------"
print " Diffusion limit info:"
print "---------------------------------------------"
print "Size in mfp of cell", mesh.getElement(0).dx * sig_t
print "Ratio of radiation energy to kinetic", Er(0.75, 0) / (rho_f(
0.25, 0) * u_f(0.0, 0)**2), GC.RAD_CONSTANT * T_inf**4 / (
rho_inf * a_inf**2)
print "Ratio of speed of light to material sound speed", GC.SPD_OF_LGT / u_f(
0.0, 0), GC.SPD_OF_LGT / (a_inf)
print "---------------------------------------------"
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s + sig_a),
ConstantCrossSection(sig_s, sig_s + sig_a))
for i in xrange(mesh.n_elems)]
# transient options
t_start = 0.0
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'double-minmod'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh=mesh,
problem_type='rad_hydro',
dt_option='constant',
dt_constant=dt_value,
slope_limiter=limiter,
time_stepper='BDF2',
use_2_cycles=True,
t_start=t_start,
t_end=t_end,
rad_BC=rad_BC,
cross_sects=cross_sects,
rad_IC=rad_IC,
hydro_IC=hydro_IC,
hydro_BC=hydro_BC,
mom_src=mom_src,
E_src=E_src,
rho_src=rho_src,
psim_src=psim_src,
psip_src=psip_src,
verbosity=verbosity,
rho_f=rho_f,
u_f=u_f,
E_f=E_f,
gamma_value=gamma_value,
cv_value=cv_value,
check_balance=True)
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
rad_exact = computeAnalyticRadSolution(mesh,
t_end,
psim=psim_f,
psip=psip_f)
#Compute error
err.append(
computeHydroL2Error(hydro_new, hydro_exact, rad_new,
rad_exact))
n_elems *= 2
dt_value *= 0.5
# compute convergence rates
rates_dx = computeHydroConvergenceRates(dx, err)
rates_dt = computeHydroConvergenceRates(dt, err)
# print convergence table
if n_cycles > 1:
printHydroConvergenceTable(dx,
err,
rates=rates_dx,
dx_desc='dx',
err_desc='$L_2$')
printHydroConvergenceTable(dt,
err,
rates=rates_dt,
dx_desc='dt',
err_desc='$L_2$')
# plot
if __name__ == '__main__':
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh,
hydro_new,
x_exact=mesh.getCellCenters(),
exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1. / GC.SPD_OF_LGT * (psim + psip)
Fr_exact_fn = (psip - psim) * RU.mu["+"]
Er_exact = []
Fr_exact = []
psip_exact = []
psim_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x': xi, 't': t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
Fr_exact.append(Fr_exact_fn.subs(substitutions))
psip_exact.append(psip_f(xi, t_end))
psim_exact.append(psim_f(xi, t_end))
plotRadErg(mesh,
rad_new.E,
Fr_edge=rad_new.F,
exact_Er=Er_exact,
exact_Fr=Fr_exact)
plotS2Erg(mesh,
rad_new.psim,
rad_new.psip,
exact_psim=psim_exact,
exact_psip=psip_exact)
plotTemperatures(mesh,
rad_new.E,
hydro_states=hydro_new,
print_values=True)
#Make a pickle to save the error tables
from sys import argv
pickname = "results/testRadHydroDiffMMS.pickle"
if len(argv) > 2:
if argv[1] == "-o":
pickname = argv[2].strip()
#Create dictionary of all the data
big_dic = {"dx": dx}
big_dic["dt"] = dt
big_dic["Errors"] = err
pickle.dump(big_dic, open(pickname, "w"))
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, alpha, c, a, mu = symbols('x t alpha c a mu')
# number of refinement cycles
n_cycles = 5
# number of elements in first cycle
n_elems = 20
# We will increase sigma and nondimensional C as we decrease the mesh
# size to ensure we stay in the diffusion limit
# numeric values
gamma_value = 5./3.
cv_value = 0.14472799784454
# run for a fixed amount of time
t_end = 2.*pi
sig_s = 0.0
#Want material speeed to be a small fraction of speed of light
#and radiation to be small relative to kinetic energy
P = 0.001 # a_r T_inf^4/(rho_inf*u_inf^2)
#Arbitrary ratio of pressure to density
alpha_value = 0.5
#Arbitrary mach number well below the sound speed. The choice of gamma and
#the cv value, as well as C and P constrain all other material reference
#parameters, but we are free to choose the material velocity below the sound
#speed to ensure no shocks are formed
M = 0.9
cfl_value = 0.6 #but 2 time steps, so really like a CFL of 0.3
dt = []
dx = []
err = []
for cycle in range(n_cycles):
#Keep ratio of and sig_a*dx constant, staying in diffusion limit
C = sig_a = 100000./20*n_elems # c/a_inf
sig_t = sig_s + sig_a
a_inf = GC.SPD_OF_LGT/C
#These lines of code assume specified C_v
T_inf = a_inf**2/(gamma_value*(gamma_value-1.)*cv_value)
rho_inf = GC.RAD_CONSTANT*T_inf**4/(P*a_inf**2)
#Specify rho_inf and determine T_inf and Cv_value
rho_inf = 1.0 #If we don't choose a reference density, then the specific heat values get ridiculous
T_inf = pow(rho_inf*P*a_inf**2/GC.RAD_CONSTANT,0.25) #to set T_inf based on rho_inf,
cv_value = a_inf**2/(T_inf*gamma_value*(gamma_value-1.)) # to set c_v, if rho specified
#Specify T_inf and solve for rho_inf and C_v value
# T_inf = 1
# rho_inf = GC.RAD_CONSTANT*T_inf**4/(P*a_inf**2)
# cv_value = a_inf**2/(T_inf*gamma_value*(gamma_value-1.)) # to set c_v, if rho specified
#Same for all approachs
p_inf = rho_inf*a_inf*a_inf
print "The dimensionalization parameters are: "
print " a_inf : ", a_inf
print " T_inf : ", T_inf
print " rho_inf : ", rho_inf
print " p_inf : ", p_inf
print " C_v : ", cv_value
print " gamma : ", gamma_value
# create solution for thermodynamic state and flow field
rho = rho_inf*(2. + sin(x-t))
u = a_inf*M*0.5*(2. + cos(x-t))
p = alpha_value*p_inf*(2. + cos(x-t))
e = p/(rho*(gamma_value-1.))
E = 0.5*rho*u*u + rho*e
# create solution for radiation field based on solution for F
# that is the leading order diffusion limit solution
T = e/cv_value
Er = a*T**4
Fr = -1./(3.*sig_t)*c*diff(Er,x) + sympify('4./3.')*Er*u
#Form psi+ and psi- from Fr and Er
psip = (Er*c*mu + Fr)/(2.*mu)
psim = (Er*c*mu - Fr)/(2.*mu)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
substitutions['a'] = GC.RAD_CONSTANT
substitutions['mu'] = RU.mu["+"]
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho*u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
T = T.subs(substitutions)
rho_f = lambdify((symbols('x'),symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'),symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'),symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'),symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'),symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'),symbols('t')), psip, "numpy")
T_f = lambdify((symbols('x'),symbols('t')), T, "numpy")
Er = Er.subs(substitutions)
Er = lambdify((symbols('x'),symbols('t')), Er, "numpy")
#For reference create lambda as ratio of aT^4/rho--u^2 to check
#P_f = lambda x,t: GC.RAD_CONSTANT*T_f(x,t)**4/(rho_f(x,t)*u_f(x,t)**2)
#dx = 1./100.
#x = -1.*dx
#t=0.2
#for i in range(100):
# x += dx
# print "ratio:", P_f(x,t)
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho = rho,
u = u,
E = E,
psim = psim,
psip = psip,
sigma_s_value = sig_s,
sigma_a_value = sig_a,
gamma_value = gamma_value,
cv_value = cv_value,
alpha_value = alpha_value,
display_equations = True)
# mesh
width = 2.*pi
mesh = Mesh(n_elems,width)
# compute hydro IC for the sake of computing initial time step size
hydro_IC = computeAnalyticHydroSolution(mesh,t=0.0,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# compute the initial time step size according to CFL conditon (actually
# half). We will then decrease this time step by the same factor as DX each
# cycle
if cycle == 0:
sound_speed = [sqrt(i.p * i.gamma / i.rho) + abs(i.u) for i in hydro_IC]
dt_vals = [cfl_value*(mesh.elements[i].dx)/sound_speed[i]
for i in xrange(len(hydro_IC))]
dt_value = min(min(dt_vals),0.5) #don't take too big of time step
print "initial dt_value", dt_value
#Adjust the end time to be an exact increment of dt_values
print "old t_end: ", t_end
print "new t_end: ", t_end
print "This cycle's dt value: ", dt_value
# create uniform mesh
mesh = Mesh(n_elems, width)
dt.append(dt_value)
dx.append(mesh.getElement(0).dx)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'periodic')
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,t=0.0,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# Dimensionless parameters. These are all evaluated at peaks of trig
# functions, very hard coded
print "---------------------------------------------"
print " Diffusion limit info:"
print "---------------------------------------------"
print "Size in mfp of cell", mesh.getElement(0).dx*sig_t
print "Ratio of radiation energy to kinetic", Er(0.75,0)/(rho_f(0.25,0)*u_f(0.0,0)**2), GC.RAD_CONSTANT*T_inf**4/(rho_inf*a_inf**2)
print "Ratio of speed of light to material sound speed", GC.SPD_OF_LGT/u_f(0.0,0), GC.SPD_OF_LGT/(a_inf)
print "---------------------------------------------"
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s+sig_a),
ConstantCrossSection(sig_s, sig_s+sig_a))
for i in xrange(mesh.n_elems)]
# transient options
t_start = 0.0
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'double-minmod'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh = mesh,
problem_type = 'rad_hydro',
dt_option = 'constant',
dt_constant = dt_value,
slope_limiter = limiter,
time_stepper = 'BDF2',
use_2_cycles = True,
t_start = t_start,
t_end = t_end,
rad_BC = rad_BC,
cross_sects = cross_sects,
rad_IC = rad_IC,
hydro_IC = hydro_IC,
hydro_BC = hydro_BC,
mom_src = mom_src,
E_src = E_src,
rho_src = rho_src,
psim_src = psim_src,
psip_src = psip_src,
verbosity = verbosity,
rho_f =rho_f,
u_f = u_f,
E_f = E_f,
gamma_value = gamma_value,
cv_value = cv_value,
check_balance = True)
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
rad_exact = computeAnalyticRadSolution(mesh, t_end,psim=psim_f,psip=psip_f)
#Compute error
err.append(computeHydroL2Error(hydro_new, hydro_exact, rad_new, rad_exact ))
n_elems *= 2
dt_value *= 0.5
# compute convergence rates
rates_dx = computeHydroConvergenceRates(dx,err)
rates_dt = computeHydroConvergenceRates(dt,err)
# print convergence table
if n_cycles > 1:
printHydroConvergenceTable(dx,err,rates=rates_dx,
dx_desc='dx',err_desc='$L_2$')
printHydroConvergenceTable(dt,err,rates=rates_dt,
dx_desc='dt',err_desc='$L_2$')
# plot
if __name__ == '__main__':
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh, hydro_new, x_exact=mesh.getCellCenters(),exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1./GC.SPD_OF_LGT*(psim + psip)
Fr_exact_fn = (psip - psim)*RU.mu["+"]
Er_exact = []
Fr_exact = []
psip_exact = []
psim_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x':xi, 't':t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
Fr_exact.append(Fr_exact_fn.subs(substitutions))
psip_exact.append(psip_f(xi,t_end))
psim_exact.append(psim_f(xi,t_end))
plotRadErg(mesh, rad_new.E, Fr_edge=rad_new.F, exact_Er=Er_exact, exact_Fr =
Fr_exact)
plotS2Erg(mesh, rad_new.psim, rad_new.psip, exact_psim=psim_exact,
exact_psip=psip_exact)
plotTemperatures(mesh, rad_new.E, hydro_states=hydro_new, print_values=True)
#Make a pickle to save the error tables
from sys import argv
pickname = "results/testRadHydroDiffMMS.pickle"
if len(argv) > 2:
if argv[1] == "-o":
pickname = argv[2].strip()
#Create dictionary of all the data
big_dic = {"dx": dx}
big_dic["dt"] = dt
big_dic["Errors"] = err
pickle.dump( big_dic, open( pickname, "w") )
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, alpha, c = symbols('x t alpha c')
#Cycles for time convergence
n_cycles = 3
# numeric values
alpha_value = 0.01
cv_value = 1.0
gamma_value = 1.4
sig_s = 1.0
sig_a = 1.0
sig_t = sig_s + sig_a
# create solution for thermodynamic state and flow field
rho = 2. + sin(2*pi*x-t)
u = 2. + cos(2*pi*x-t)
p = 0.5*(2. + cos(2*pi*x-t))
e = p/(rho*(gamma_value-1.))
E = 0.5*rho*u*u + rho*e
rho = sympify('2')*sin(t/4.)+2.
u = sympify('3')*sin(t/4.)+3.
E = sympify('10')*sin(t/4.)+10.
# create solution for radiation field based on solution for F
# that is the leading order diffusion limit solution
a = GC.RAD_CONSTANT
c = GC.SPD_OF_LGT
mu = RU.mu["+"]
#Equilibrium diffusion solution
Er = (2.+cos(2*pi*x-t))
Fr = (2.+cos(2*pi*x-t))*c
psip = (Er*c*mu + Fr)/(2.*mu)
psim = (Er*c*mu - Fr)/(2.*mu)
psip = sympify('10')*c+1*sin(t/4.)
psim = sympify('10')*c+1*cos(t/4.)
#Form psi+ and psi- from Fr and Er
#psip = sympify('5.')*c
#psim = sympify('5.')*c
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho = rho,
u = u,
E = E,
psim = psim,
psip = psip,
sigma_s_value = sig_s,
sigma_a_value = sig_a,
gamma_value = gamma_value,
cv_value = cv_value,
alpha_value = alpha_value,
display_equations = True)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho*u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
rho_f = lambdify((symbols('x'),symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'),symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'),symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'),symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'),symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'),symbols('t')), psip, "numpy")
dt_value = 0.0001
dt = []
err = []
#Loop over cycles for time convergence
for cycle in range(n_cycles):
# create uniform mesh
n_elems = 50
width = 1.0
mesh = Mesh(n_elems, width)
#Store
dt.append(dt_value)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'periodic')
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,t=0.0,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s+sig_a),
ConstantCrossSection(sig_s, sig_s+sig_a))
for i in xrange(mesh.n_elems)]
# transient options
t_start = 0.0
t_end = 0.001
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'none'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh = mesh,
problem_type = 'rad_hydro',
dt_option = 'constant',
dt_constant = dt_value,
slope_limiter = limiter,
use_2_cycles = True,
t_start = t_start,
t_end = t_end,
rad_BC = rad_BC,
cross_sects = cross_sects,
rad_IC = rad_IC,
hydro_IC = hydro_IC,
hydro_BC = hydro_BC,
mom_src = mom_src,
E_src = E_src,
rho_src = rho_src,
psim_src = psim_src,
psip_src = psip_src,
verbosity = verbosity,
rho_f =rho_f,
u_f = u_f,
E_f = E_f,
gamma_value = gamma_value,
cv_value = cv_value,
check_balance = True)
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
#Compute error
err.append(computeHydroL2Error(hydro_new, hydro_exact))
dt_value /= 2.
# compute convergence rates
rates = computeHydroConvergenceRates(dt,err)
# plot
if __name__ == '__main__':
# plot radiation solution
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# print convergence table
if n_cycles > 1:
printHydroConvergenceTable(dt,err,rates=rates,
dx_desc='dt',err_desc='L2')
# plot hydro solution
plotHydroSolutions(mesh, hydro_new, x_exact=mesh.getCellCenters(),
exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1./GC.SPD_OF_LGT*(psim + psip)
Fr_exact_fn = (psip - psim)*RU.mu["+"]
Er_exact = []
Fr_exact = []
psip_exact = []
psim_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x':xi, 't':t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
Fr_exact.append(Fr_exact_fn.subs(substitutions))
psip_exact.append(psip_f(xi,t_end))
psim_exact.append(psim_f(xi,t_end))
plotRadErg(mesh, rad_new.E, Fr_edge=rad_new.F, exact_Er=Er_exact, exact_Fr =
Fr_exact)
plotRadErg(mesh, rad_new.psim, rad_new.psip, exact_Er=psip_exact,
exact_Fr=psim_exact)
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, A, B, C, c, cv, gamma, mu, alpha = \
symbols('x t A B C c cv gamma mu alpha')
#These constants, as well as cross sections and C_v, rho_ref
#will set the material
#and radiation to be small relative to kinetic energy
Ctilde = 100.
P = 0.1
#Arbitrary mach number well below the sound speed. The choice of gamma and
#the cv value, as well as C and P constrain all other material reference
#parameters, but we are free to choose the material velocity below the sound
#speed to ensure no shocks are formed
M = 0.9
cfl_value = 0.6 #but 2 time steps, so really like a CFL of 0.3
width = 2.0*math.pi
# numeric values
gamma_value = 5./3.
cv_value = 0.14472799784454
# number of refinement cycles
n_cycles = 5
# number of elements in first cycle
n_elems = 40
# transient options
t_start = 0.0
t_end = 2.0*math.pi/Ctilde
# numeric values
A_value = 1.0
B_value = 1.0
C_value = 1.0
alpha_value = 0.5
gamma_value = 5.0/3.0
sig_s_value = 0.0
sig_a_value = 1.0
# numeric values
a_inf = GC.SPD_OF_LGT/Ctilde
#T_inf = a_inf**2/(gamma_value*(gamma_value - 1.)*cv_value)
#rho_inf = GC.RAD_CONSTANT*T_inf**4/(P*a_inf**2)
rho_inf = 1.0
T_inf = pow(rho_inf*P*a_inf**2/GC.RAD_CONSTANT,0.25) #to set T_inf based on rho_inf,
cv_value = a_inf**2/(T_inf*gamma_value*(gamma_value-1.)) # to set c_v, if rho specified
p_inf = rho_inf*a_inf*a_inf
Er_inf = GC.RAD_CONSTANT*T_inf**4
# MMS solutions
rho = rho_inf*A*(sin(B*x-C*t)+2.)
u = M*a_inf*1./(A*(sin(B*x-C*t)+2.))
p = p_inf*A*alpha*(sin(B*x-C*t)+2.)
Er = alpha*(sin(B*x-Ctilde*C*t)+2.)*Er_inf
Fr = alpha*(sin(B*x-Ctilde*C*t)+2.)*c*Er_inf
#Er = 0.5*(sin(2*pi*x - 10.*t) + 2.)/c
#Fr = 0.5*(sin(2*pi*x - 10.*t) + 2.)
e = p/(rho*(gamma_value-1.))
E = 0.5*rho*u*u + rho*e
print "The dimensionalization parameters are: "
print " a_inf : ", a_inf
print " T_inf : ", T_inf
print " rho_inf : ", rho_inf
print " p_inf : ", p_inf
# derived solutions
T = e/cv_value
E = rho*(u*u/2 + e)
psip = (Er*c + Fr/mu)/2
psim = (Er*c - Fr/mu)/2
# create list of substitutions
substitutions = dict()
substitutions['A'] = A_value
substitutions['B'] = B_value
substitutions['C'] = C_value
substitutions['c'] = GC.SPD_OF_LGT
substitutions['cv'] = cv_value
substitutions['gamma'] = gamma_value
substitutions['mu'] = RU.mu["+"]
substitutions['alpha'] = alpha_value
# make substitutions
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho*u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho = rho,
u = u,
E = E,
psim = psim,
psip = psip,
sigma_s_value = sig_s_value,
sigma_a_value = sig_a_value,
gamma_value = gamma_value,
cv_value = cv_value,
alpha_value = alpha_value,
display_equations = True)
# create functions for exact solutions
rho_f = lambdify((symbols('x'),symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'),symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'),symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'),symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'),symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'),symbols('t')), psip, "numpy")
dt = []
dx = []
err = []
for cycle in range(n_cycles):
# create uniform mesh
mesh = Mesh(n_elems, width)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'periodic')
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,t=0.0,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s_value, sig_s_value+sig_a_value),
ConstantCrossSection(sig_s_value, sig_s_value+sig_a_value))
for i in xrange(mesh.n_elems)]
# compute the initial time step size according to CFL conditon (actually
# half). We will then decrease this time step by the same factor as DX each
# cycle
if cycle == 0:
sound_speed = [sqrt(i.p * i.gamma / i.rho) + abs(i.u) for i in hydro_IC]
dt_vals = [cfl_value*(mesh.elements[i].dx)/sound_speed[i]
for i in xrange(len(hydro_IC))]
dt_value = min(dt_vals)
print "dt_value for hydro: ", dt_value
#Make sure not taking too large of step for radiation time scale
period = 2.*math.pi/Ctilde
dt_value = min(dt_value, period/8.)
print "initial dt_value", dt_value
#Adjust the end time to be an exact increment of dt_values
print "t_end: ", t_end
print "This cycle's dt value: ", dt_value
dt.append(dt_value)
dx.append(mesh.getElement(0).dx)
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'double-minmod'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh = mesh,
problem_type = 'rad_hydro',
dt_option = 'constant',
dt_constant = dt_value,
slope_limiter = limiter,
time_stepper = 'BDF2',
use_2_cycles = True,
t_start = t_start,
t_end = t_end,
rad_BC = rad_BC,
cross_sects = cross_sects,
rad_IC = rad_IC,
hydro_IC = hydro_IC,
hydro_BC = hydro_BC,
mom_src = mom_src,
E_src = E_src,
rho_src = rho_src,
psim_src = psim_src,
psip_src = psip_src,
verbosity = verbosity,
rho_f = rho_f,
u_f = u_f,
E_f = E_f,
gamma_value = gamma_value,
cv_value = cv_value,
check_balance = False)
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
rad_exact = computeAnalyticRadSolution(mesh, t_end,psim=psim_f,psip=psip_f)
#Compute error
err.append(computeHydroL2Error(hydro_new, hydro_exact, rad_new, rad_exact ))
n_elems *= 2
dt_value *= 0.5
# compute convergence rates
rates_dx = computeHydroConvergenceRates(dx,err)
rates_dt = computeHydroConvergenceRates(dt,err)
# print convergence table
if n_cycles > 1:
printHydroConvergenceTable(dx,err,rates=rates_dx,
dx_desc='dx',err_desc='$L_2$')
printHydroConvergenceTable(dt,err,rates=rates_dt,
dx_desc='dt',err_desc='$L_2$')
# plot
if __name__ == '__main__':
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh, hydro_new, x_exact=mesh.getCellCenters(),exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1./GC.SPD_OF_LGT*(psim + psip)
Fr_exact_fn = (psip - psim)*RU.mu["+"]
Er_exact = []
Fr_exact = []
psip_exact = []
psim_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x':xi, 't':t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
Fr_exact.append(Fr_exact_fn.subs(substitutions))
psip_exact.append(psip_f(xi,t_end))
psim_exact.append(psim_f(xi,t_end))
plotRadErg(mesh, rad_new.E, Fr_edge=rad_new.F, exact_Er=Er_exact, exact_Fr =
Fr_exact)
plotS2Erg(mesh, rad_new.psim, rad_new.psip, exact_psim=psim_exact,
exact_psip=psip_exact)
plotTemperatures(mesh, rad_new.E, hydro_states=hydro_new, print_values=True)
#Make a pickle to save the error tables
from sys import argv
pickname = "results/testRadHydroStreamingMMSC100.pickle"
if len(argv) > 2:
if argv[1] == "-o":
pickname = argv[2].strip()
#Create dictionary of all the data
big_dic = {"dx": dx}
big_dic["dt"] = dt
big_dic["Errors"] = err
pickle.dump( big_dic, open( pickname, "w") )
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, alpha, c = symbols('x t alpha c')
# numeric values
alpha_value = 0.01
cv_value = 1.0
gamma_value = 1.4
sig_s = 1.0
sig_a = 1.0
# create solution for thermodynamic state and flow field
# rho = sympify('4.0')
# u = sympify('1.0')
# E = sympify('10.0')
rho = 2. + sin(2*pi*x+t)
u = 2. + cos(2*pi*x-t)
p = 2. + cos(2*pi*x+t)
e = p/(rho*(gamma_value-1.))
E = 0.5*rho*u*u + rho*e
# create solution for radiation field
rad_scale = 50*c
psim = rad_scale*(2*t*sin(pi*(1-x))+2+0.1*t)
psip = rad_scale*(t*sin(pi*x)+2+0.1*t)
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho = rho,
u = u,
E = E,
psim = psim,
psip = psip,
sigma_s_value = sig_s,
sigma_a_value = sig_a,
gamma_value = gamma_value,
cv_value = cv_value,
alpha_value = alpha_value,
display_equations = True)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho*u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
rho_f = lambdify((symbols('x'),symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'),symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'),symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'),symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'),symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'),symbols('t')), psip, "numpy")
# create uniform mesh
n_elems = 50
width = 1.0
mesh = Mesh(n_elems, width)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'dirichlet',psip_BC=psip_f,psim_BC=psim_f)
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,t=0.0,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s+sig_a),
ConstantCrossSection(sig_s, sig_s+sig_a))
for i in xrange(mesh.n_elems)]
# transient options
t_start = 0.0
# t_end = 0.005
t_end = 0.02
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'none'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh = mesh,
problem_type = 'rad_hydro',
dt_option = 'CFL',
CFL = 0.5,
# dt_option = 'constant',
# dt_constant = 0.0002,
slope_limiter = limiter,
time_stepper = 'BDF2',
use_2_cycles = True,
t_start = t_start,
t_end = t_end,
rad_BC = rad_BC,
cross_sects = cross_sects,
rad_IC = rad_IC,
hydro_IC = hydro_IC,
hydro_BC = hydro_BC,
mom_src = mom_src,
E_src = E_src,
rho_src = rho_src,
psim_src = psim_src,
psip_src = psip_src,
verbosity = verbosity,
rho_f =rho_f,
u_f = u_f,
E_f = E_f,
gamma_value = gamma_value,
cv_value = cv_value,
check_balance = True)
# plot
if __name__ == '__main__':
# plot radiation solution
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh, hydro_new, x_exact=mesh.getCellCenters(),exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1./GC.SPD_OF_LGT*(psim + psip)
Er_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x':xi, 't':t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
plotRadErg(mesh, rad_new.E, exact_Er=Er_exact)
def test_RadHydroMMS(self):
# slope limiter: choices are:
# none step minmod double-minmod superbee minbee vanleer
slope_limiter = 'none'
# number of elements
n_elems = 50
# end time
t_end = 0.1
# choice of solutions for hydro
hydro_case = "linear" # constant linear exponential
# choice of solutions for radiation
rad_case = "zero" # zero constant sin
# declare symbolic variables
x, t, alpha, c = symbols('x t alpha c')
# create solution for thermodynamic state and flow field
if hydro_case == "constant":
rho = sympify('4.0')
u = sympify('1.2')
E = sympify('10.0')
elif hydro_case == "linear":
rho = 1 + x - t
u = sympify('1')
E = 5 + 5 * (x - 0.5)**2
elif hydro_case == "exponential":
rho = exp(x + t) + 5
u = exp(-x) * sin(t) - 1
E = 10 * exp(x + t)
else:
raise NotImplementedError("Invalid hydro test case")
# create solution for radiation field
if rad_case == "zero":
psim = sympify('0')
psip = sympify('0')
elif rad_case == "constant":
psim = 50 * c
psip = 50 * c
elif rad_case == "sin":
rad_scale = 50 * c
psim = rad_scale * 2 * t * sin(pi * (1 - x)) + 10 * c
psip = rad_scale * t * sin(pi * x) + 10 * c
else:
raise NotImplementedError("Invalid radiation test case")
# numeric values
alpha_value = 0.01
cv_value = 1.0
gamma_value = 1.4
sig_s = 1.0
sig_a = 1.0
#sig_a = 0.0
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho=rho,
u=u,
E=E,
psim=psim,
psip=psip,
sigma_s_value=sig_s,
sigma_a_value=sig_a,
gamma_value=gamma_value,
cv_value=cv_value,
alpha_value=alpha_value,
display_equations=False)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho * u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
rho_f = lambdify((symbols('x'), symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'), symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'), symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'), symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'), symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'), symbols('t')), psip, "numpy")
# create uniform mesh
width = 1.0
mesh = Mesh(n_elems, width)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# compute radiation BC; assumes BC is independent of time
psi_left = psip_f(x=0.0, t=0.0)
psi_right = psim_f(x=width, t=0.0)
#Create Radiation BC object
rad_BC = RadBC(mesh,
"dirichlet",
psi_left=psi_left,
psi_right=psi_right)
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,
t=0.0,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='dirichlet',
mesh=mesh,
rho_BC=rho_f,
mom_BC=mom_f,
erg_BC=E_f)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s + sig_a),
ConstantCrossSection(sig_s, sig_s + sig_a))
for i in xrange(mesh.n_elems)]
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh=mesh,
problem_type='rad_hydro',
dt_option='CFL',
#dt_option = 'constant',
CFL=0.5,
#dt_constant = 0.002,
slope_limiter=slope_limiter,
time_stepper='BDF2',
use_2_cycles=True,
t_start=0.0,
t_end=t_end,
rad_BC=rad_BC,
cross_sects=cross_sects,
rad_IC=rad_IC,
hydro_IC=hydro_IC,
hydro_BC=hydro_BC,
mom_src=mom_src,
E_src=E_src,
rho_src=rho_src,
psim_src=psim_src,
psip_src=psip_src,
verbosity=verbosity,
check_balance=False)
# plot
if __name__ == '__main__':
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh,
hydro_new,
x_exact=mesh.getCellCenters(),
exact=hydro_exact)
# compute exact radiation energy
Er_exact_fn = 1. / GC.SPD_OF_LGT * (psim + psip)
Er_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x': xi, 't': t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
# plot radiation energy
plotRadErg(mesh, rad_new.E, exact_Er=Er_exact)
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, A, B, C, c, cv, gamma, mu, alpha = \
symbols('x t A B C c cv gamma mu alpha')
#These constants, as well as cross sections and C_v, rho_ref
#will set the material
#and radiation to be small relative to kinetic energy
Ctilde = 100.
P = 0.1
#Arbitrary mach number well below the sound speed. The choice of gamma and
#the cv value, as well as C and P constrain all other material reference
#parameters, but we are free to choose the material velocity below the sound
#speed to ensure no shocks are formed
M = 0.9
cfl_value = 0.6 #but 2 time steps, so really like a CFL of 0.3
width = 2.0 * math.pi
# numeric values
gamma_value = 5. / 3.
cv_value = 0.14472799784454
# number of refinement cycles
n_cycles = 5
# number of elements in first cycle
n_elems = 40
# transient options
t_start = 0.0
t_end = 2.0 * math.pi / Ctilde
# numeric values
A_value = 1.0
B_value = 1.0
C_value = 1.0
alpha_value = 0.5
gamma_value = 5.0 / 3.0
sig_s_value = 0.0
sig_a_value = 1.0
# numeric values
a_inf = GC.SPD_OF_LGT / Ctilde
#T_inf = a_inf**2/(gamma_value*(gamma_value - 1.)*cv_value)
#rho_inf = GC.RAD_CONSTANT*T_inf**4/(P*a_inf**2)
rho_inf = 1.0
T_inf = pow(rho_inf * P * a_inf**2 / GC.RAD_CONSTANT,
0.25) #to set T_inf based on rho_inf,
cv_value = a_inf**2 / (T_inf * gamma_value * (gamma_value - 1.)
) # to set c_v, if rho specified
p_inf = rho_inf * a_inf * a_inf
Er_inf = GC.RAD_CONSTANT * T_inf**4
# MMS solutions
rho = rho_inf * A * (sin(B * x - C * t) + 2.)
u = M * a_inf * 1. / (A * (sin(B * x - C * t) + 2.))
p = p_inf * A * alpha * (sin(B * x - C * t) + 2.)
Er = alpha * (sin(B * x - Ctilde * C * t) + 2.) * Er_inf
Fr = alpha * (sin(B * x - Ctilde * C * t) + 2.) * c * Er_inf
#Er = 0.5*(sin(2*pi*x - 10.*t) + 2.)/c
#Fr = 0.5*(sin(2*pi*x - 10.*t) + 2.)
e = p / (rho * (gamma_value - 1.))
E = 0.5 * rho * u * u + rho * e
print "The dimensionalization parameters are: "
print " a_inf : ", a_inf
print " T_inf : ", T_inf
print " rho_inf : ", rho_inf
print " p_inf : ", p_inf
# derived solutions
T = e / cv_value
E = rho * (u * u / 2 + e)
psip = (Er * c + Fr / mu) / 2
psim = (Er * c - Fr / mu) / 2
# create list of substitutions
substitutions = dict()
substitutions['A'] = A_value
substitutions['B'] = B_value
substitutions['C'] = C_value
substitutions['c'] = GC.SPD_OF_LGT
substitutions['cv'] = cv_value
substitutions['gamma'] = gamma_value
substitutions['mu'] = RU.mu["+"]
substitutions['alpha'] = alpha_value
# make substitutions
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho * u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho=rho,
u=u,
E=E,
psim=psim,
psip=psip,
sigma_s_value=sig_s_value,
sigma_a_value=sig_a_value,
gamma_value=gamma_value,
cv_value=cv_value,
alpha_value=alpha_value,
display_equations=True)
# create functions for exact solutions
rho_f = lambdify((symbols('x'), symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'), symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'), symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'), symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'), symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'), symbols('t')), psip, "numpy")
dt = []
dx = []
err = []
for cycle in range(n_cycles):
# create uniform mesh
mesh = Mesh(n_elems, width)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'periodic')
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,
t=0.0,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s_value,
sig_s_value + sig_a_value),
ConstantCrossSection(sig_s_value,
sig_s_value + sig_a_value))
for i in xrange(mesh.n_elems)]
# compute the initial time step size according to CFL conditon (actually
# half). We will then decrease this time step by the same factor as DX each
# cycle
if cycle == 0:
sound_speed = [
sqrt(i.p * i.gamma / i.rho) + abs(i.u) for i in hydro_IC
]
dt_vals = [
cfl_value * (mesh.elements[i].dx) / sound_speed[i]
for i in xrange(len(hydro_IC))
]
dt_value = min(dt_vals)
print "dt_value for hydro: ", dt_value
#Make sure not taking too large of step for radiation time scale
period = 2. * math.pi / Ctilde
dt_value = min(dt_value, period / 8.)
print "initial dt_value", dt_value
#Adjust the end time to be an exact increment of dt_values
print "t_end: ", t_end
print "This cycle's dt value: ", dt_value
dt.append(dt_value)
dx.append(mesh.getElement(0).dx)
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'double-minmod'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh=mesh,
problem_type='rad_hydro',
dt_option='constant',
dt_constant=dt_value,
slope_limiter=limiter,
time_stepper='BDF2',
use_2_cycles=True,
t_start=t_start,
t_end=t_end,
rad_BC=rad_BC,
cross_sects=cross_sects,
rad_IC=rad_IC,
hydro_IC=hydro_IC,
hydro_BC=hydro_BC,
mom_src=mom_src,
E_src=E_src,
rho_src=rho_src,
psim_src=psim_src,
psip_src=psip_src,
verbosity=verbosity,
rho_f=rho_f,
u_f=u_f,
E_f=E_f,
gamma_value=gamma_value,
cv_value=cv_value,
check_balance=False)
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
rad_exact = computeAnalyticRadSolution(mesh,
t_end,
psim=psim_f,
psip=psip_f)
#Compute error
err.append(
computeHydroL2Error(hydro_new, hydro_exact, rad_new,
rad_exact))
n_elems *= 2
dt_value *= 0.5
# compute convergence rates
rates_dx = computeHydroConvergenceRates(dx, err)
rates_dt = computeHydroConvergenceRates(dt, err)
# print convergence table
if n_cycles > 1:
printHydroConvergenceTable(dx,
err,
rates=rates_dx,
dx_desc='dx',
err_desc='$L_2$')
printHydroConvergenceTable(dt,
err,
rates=rates_dt,
dx_desc='dt',
err_desc='$L_2$')
# plot
if __name__ == '__main__':
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh,
hydro_new,
x_exact=mesh.getCellCenters(),
exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1. / GC.SPD_OF_LGT * (psim + psip)
Fr_exact_fn = (psip - psim) * RU.mu["+"]
Er_exact = []
Fr_exact = []
psip_exact = []
psim_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x': xi, 't': t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
Fr_exact.append(Fr_exact_fn.subs(substitutions))
psip_exact.append(psip_f(xi, t_end))
psim_exact.append(psim_f(xi, t_end))
plotRadErg(mesh,
rad_new.E,
Fr_edge=rad_new.F,
exact_Er=Er_exact,
exact_Fr=Fr_exact)
plotS2Erg(mesh,
rad_new.psim,
rad_new.psip,
exact_psim=psim_exact,
exact_psip=psip_exact)
plotTemperatures(mesh,
rad_new.E,
hydro_states=hydro_new,
print_values=True)
#Make a pickle to save the error tables
from sys import argv
pickname = "results/testRadHydroStreamingMMSC100.pickle"
if len(argv) > 2:
if argv[1] == "-o":
pickname = argv[2].strip()
#Create dictionary of all the data
big_dic = {"dx": dx}
big_dic["dt"] = dt
big_dic["Errors"] = err
pickle.dump(big_dic, open(pickname, "w"))
def test_RadHydroMMS(self):
# slope limiter: choices are:
# none step minmod double-minmod superbee minbee vanleer
slope_limiter = 'none'
# number of elements
n_elems = 50
# end time
t_end = 0.1
# choice of solutions for hydro
hydro_case = "linear" # constant linear exponential
# choice of solutions for radiation
rad_case = "zero" # zero constant sin
# declare symbolic variables
x, t, alpha, c = symbols('x t alpha c')
# create solution for thermodynamic state and flow field
if hydro_case == "constant":
rho = sympify('4.0')
u = sympify('1.2')
E = sympify('10.0')
elif hydro_case == "linear":
rho = 1 + x - t
u = sympify('1')
E = 5 + 5*(x - 0.5)**2
elif hydro_case == "exponential":
rho = exp(x+t)+5
u = exp(-x)*sin(t) - 1
E = 10*exp(x+t)
else:
raise NotImplementedError("Invalid hydro test case")
# create solution for radiation field
if rad_case == "zero":
psim = sympify('0')
psip = sympify('0')
elif rad_case == "constant":
psim = 50*c
psip = 50*c
elif rad_case == "sin":
rad_scale = 50*c
psim = rad_scale*2*t*sin(pi*(1-x))+10*c
psip = rad_scale*t*sin(pi*x)+10*c
else:
raise NotImplementedError("Invalid radiation test case")
# numeric values
alpha_value = 0.01
cv_value = 1.0
gamma_value = 1.4
sig_s = 1.0
sig_a = 1.0
#sig_a = 0.0
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho = rho,
u = u,
E = E,
psim = psim,
psip = psip,
sigma_s_value = sig_s,
sigma_a_value = sig_a,
gamma_value = gamma_value,
cv_value = cv_value,
alpha_value = alpha_value,
display_equations = False)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho*u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
rho_f = lambdify((symbols('x'),symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'),symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'),symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'),symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'),symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'),symbols('t')), psip, "numpy")
# create uniform mesh
width = 1.0
mesh = Mesh(n_elems, width)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# compute radiation BC; assumes BC is independent of time
psi_left = psip_f(x=0.0, t=0.0)
psi_right = psim_f(x=width, t=0.0)
#Create Radiation BC object
rad_BC = RadBC(mesh, "dirichlet", psi_left=psi_left, psi_right=psi_right)
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,t=0.0,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='dirichlet', mesh=mesh, rho_BC=rho_f,
mom_BC=mom_f, erg_BC=E_f)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s+sig_a),
ConstantCrossSection(sig_s, sig_s+sig_a))
for i in xrange(mesh.n_elems)]
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh = mesh,
problem_type = 'rad_hydro',
dt_option = 'CFL',
#dt_option = 'constant',
CFL = 0.5,
#dt_constant = 0.002,
slope_limiter = slope_limiter,
time_stepper = 'BDF2',
use_2_cycles = True,
t_start = 0.0,
t_end = t_end,
rad_BC = rad_BC,
cross_sects = cross_sects,
rad_IC = rad_IC,
hydro_IC = hydro_IC,
hydro_BC = hydro_BC,
mom_src = mom_src,
E_src = E_src,
rho_src = rho_src,
psim_src = psim_src,
psip_src = psip_src,
verbosity = verbosity,
check_balance = False)
# plot
if __name__ == '__main__':
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh, t=t_end,
rho=rho_f, u=u_f, E=E_f, cv=cv_value, gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh, hydro_new, x_exact=mesh.getCellCenters(),
exact=hydro_exact)
# compute exact radiation energy
Er_exact_fn = 1./GC.SPD_OF_LGT*(psim + psip)
Er_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x':xi, 't':t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
# plot radiation energy
plotRadErg(mesh, rad_new.E, exact_Er=Er_exact)
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, alpha, c = symbols('x t alpha c')
# numeric values
alpha_value = 0.01
cv_value = 1.0
gamma_value = 1.4
sig_s = 1.0
sig_a = 1.0
# create solution for thermodynamic state and flow field
# rho = sympify('4.0')
# u = sympify('1.0')
# E = sympify('10.0')
rho = 2. + sin(2 * pi * x + t)
u = 2. + cos(2 * pi * x - t)
p = 2. + cos(2 * pi * x + t)
e = p / (rho * (gamma_value - 1.))
E = 0.5 * rho * u * u + rho * e
# create solution for radiation field
rad_scale = 50 * c
psim = rad_scale * (2 * t * sin(pi * (1 - x)) + 2 + 0.1 * t)
psip = rad_scale * (t * sin(pi * x) + 2 + 0.1 * t)
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho=rho,
u=u,
E=E,
psim=psim,
psip=psip,
sigma_s_value=sig_s,
sigma_a_value=sig_a,
gamma_value=gamma_value,
cv_value=cv_value,
alpha_value=alpha_value,
display_equations=True)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho * u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
rho_f = lambdify((symbols('x'), symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'), symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'), symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'), symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'), symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'), symbols('t')), psip, "numpy")
# create uniform mesh
n_elems = 50
width = 1.0
mesh = Mesh(n_elems, width)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'dirichlet', psip_BC=psip_f, psim_BC=psim_f)
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,
t=0.0,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s + sig_a),
ConstantCrossSection(sig_s, sig_s + sig_a))
for i in xrange(mesh.n_elems)]
# transient options
t_start = 0.0
# t_end = 0.005
t_end = 0.02
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'none'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh=mesh,
problem_type='rad_hydro',
dt_option='CFL',
CFL=0.5,
# dt_option = 'constant',
# dt_constant = 0.0002,
slope_limiter=limiter,
time_stepper='BDF2',
use_2_cycles=True,
t_start=t_start,
t_end=t_end,
rad_BC=rad_BC,
cross_sects=cross_sects,
rad_IC=rad_IC,
hydro_IC=hydro_IC,
hydro_BC=hydro_BC,
mom_src=mom_src,
E_src=E_src,
rho_src=rho_src,
psim_src=psim_src,
psip_src=psip_src,
verbosity=verbosity,
rho_f=rho_f,
u_f=u_f,
E_f=E_f,
gamma_value=gamma_value,
cv_value=cv_value,
check_balance=True)
# plot
if __name__ == '__main__':
# plot radiation solution
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# plot hydro solution
plotHydroSolutions(mesh,
hydro_new,
x_exact=mesh.getCellCenters(),
exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1. / GC.SPD_OF_LGT * (psim + psip)
Er_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x': xi, 't': t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
plotRadErg(mesh, rad_new.E, exact_Er=Er_exact)
def test_RadHydroMMS(self):
# declare symbolic variables
x, t, alpha, c = symbols('x t alpha c')
#Cycles for time convergence
n_cycles = 3
# numeric values
alpha_value = 0.01
cv_value = 1.0
gamma_value = 1.4
sig_s = 1.0
sig_a = 1.0
sig_t = sig_s + sig_a
# create solution for thermodynamic state and flow field
rho = 2. + sin(2 * pi * x - t)
u = 2. + cos(2 * pi * x - t)
p = 0.5 * (2. + cos(2 * pi * x - t))
e = p / (rho * (gamma_value - 1.))
E = 0.5 * rho * u * u + rho * e
rho = sympify('2') * sin(t / 4.) + 2.
u = sympify('3') * sin(t / 4.) + 3.
E = sympify('10') * sin(t / 4.) + 10.
# create solution for radiation field based on solution for F
# that is the leading order diffusion limit solution
a = GC.RAD_CONSTANT
c = GC.SPD_OF_LGT
mu = RU.mu["+"]
#Equilibrium diffusion solution
Er = (2. + cos(2 * pi * x - t))
Fr = (2. + cos(2 * pi * x - t)) * c
psip = (Er * c * mu + Fr) / (2. * mu)
psim = (Er * c * mu - Fr) / (2. * mu)
psip = sympify('10') * c + 1 * sin(t / 4.)
psim = sympify('10') * c + 1 * cos(t / 4.)
#Form psi+ and psi- from Fr and Er
#psip = sympify('5.')*c
#psim = sympify('5.')*c
# create MMS source functions
rho_src, mom_src, E_src, psim_src, psip_src = createMMSSourceFunctionsRadHydro(
rho=rho,
u=u,
E=E,
psim=psim,
psip=psip,
sigma_s_value=sig_s,
sigma_a_value=sig_a,
gamma_value=gamma_value,
cv_value=cv_value,
alpha_value=alpha_value,
display_equations=True)
# create functions for exact solutions
substitutions = dict()
substitutions['alpha'] = alpha_value
substitutions['c'] = GC.SPD_OF_LGT
rho = rho.subs(substitutions)
u = u.subs(substitutions)
mom = rho * u
E = E.subs(substitutions)
psim = psim.subs(substitutions)
psip = psip.subs(substitutions)
rho_f = lambdify((symbols('x'), symbols('t')), rho, "numpy")
u_f = lambdify((symbols('x'), symbols('t')), u, "numpy")
mom_f = lambdify((symbols('x'), symbols('t')), mom, "numpy")
E_f = lambdify((symbols('x'), symbols('t')), E, "numpy")
psim_f = lambdify((symbols('x'), symbols('t')), psim, "numpy")
psip_f = lambdify((symbols('x'), symbols('t')), psip, "numpy")
dt_value = 0.0001
dt = []
err = []
#Loop over cycles for time convergence
for cycle in range(n_cycles):
# create uniform mesh
n_elems = 50
width = 1.0
mesh = Mesh(n_elems, width)
#Store
dt.append(dt_value)
# compute radiation IC
psi_IC = computeRadiationVector(psim_f, psip_f, mesh, t=0.0)
rad_IC = Radiation(psi_IC)
# create rad BC object
rad_BC = RadBC(mesh, 'periodic')
# compute hydro IC
hydro_IC = computeAnalyticHydroSolution(mesh,
t=0.0,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# create hydro BC
hydro_BC = HydroBC(bc_type='periodic', mesh=mesh)
# create cross sections
cross_sects = [(ConstantCrossSection(sig_s, sig_s + sig_a),
ConstantCrossSection(sig_s, sig_s + sig_a))
for i in xrange(mesh.n_elems)]
# transient options
t_start = 0.0
t_end = 0.001
# if run standalone, then be verbose
if __name__ == '__main__':
verbosity = 2
else:
verbosity = 0
#slope limiter
limiter = 'none'
# run the rad-hydro transient
rad_new, hydro_new = runNonlinearTransient(
mesh=mesh,
problem_type='rad_hydro',
dt_option='constant',
dt_constant=dt_value,
slope_limiter=limiter,
use_2_cycles=True,
t_start=t_start,
t_end=t_end,
rad_BC=rad_BC,
cross_sects=cross_sects,
rad_IC=rad_IC,
hydro_IC=hydro_IC,
hydro_BC=hydro_BC,
mom_src=mom_src,
E_src=E_src,
rho_src=rho_src,
psim_src=psim_src,
psip_src=psip_src,
verbosity=verbosity,
rho_f=rho_f,
u_f=u_f,
E_f=E_f,
gamma_value=gamma_value,
cv_value=cv_value,
check_balance=True)
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
#Compute error
err.append(computeHydroL2Error(hydro_new, hydro_exact))
dt_value /= 2.
# compute convergence rates
rates = computeHydroConvergenceRates(dt, err)
# plot
if __name__ == '__main__':
# plot radiation solution
# compute exact hydro solution
hydro_exact = computeAnalyticHydroSolution(mesh,
t=t_end,
rho=rho_f,
u=u_f,
E=E_f,
cv=cv_value,
gamma=gamma_value)
# print convergence table
if n_cycles > 1:
printHydroConvergenceTable(dt,
err,
rates=rates,
dx_desc='dt',
err_desc='L2')
# plot hydro solution
plotHydroSolutions(mesh,
hydro_new,
x_exact=mesh.getCellCenters(),
exact=hydro_exact)
#plot exact and our E_r
Er_exact_fn = 1. / GC.SPD_OF_LGT * (psim + psip)
Fr_exact_fn = (psip - psim) * RU.mu["+"]
Er_exact = []
Fr_exact = []
psip_exact = []
psim_exact = []
x = mesh.getCellCenters()
for xi in x:
substitutions = {'x': xi, 't': t_end}
Er_exact.append(Er_exact_fn.subs(substitutions))
Fr_exact.append(Fr_exact_fn.subs(substitutions))
psip_exact.append(psip_f(xi, t_end))
psim_exact.append(psim_f(xi, t_end))
plotRadErg(mesh,
rad_new.E,
Fr_edge=rad_new.F,
exact_Er=Er_exact,
exact_Fr=Fr_exact)
plotRadErg(mesh,
rad_new.psim,
rad_new.psip,
exact_Er=psip_exact,
exact_Fr=psim_exact)