def verify_timestep_order(stepper_getter, order, setup=None, dtmul=1):
eocrec = EOCRecorder()
for n in range(4,9):
dt = 2**(-n) * dtmul
stepper = stepper_getter()
if setup is not None:
setup(stepper, dt)
error = get_error(stepper,dt)
eocrec.add_data_point(1/dt, error)
print "------------------------------------------------------"
print "ORDER %d, %s" % (order, stepper)
print "------------------------------------------------------"
print eocrec.pretty_print()
orderest = eocrec.estimate_order_of_convergence()[0,1]
#print orderest, order
assert orderest > order*0.95
def __call__(self):
from hedge.tools import EOCRecorder
eocrec = EOCRecorder()
for n in range(4,7):
dt = 2**(-n)
from hedge.timestep.multirate_ab import TwoRateAdamsBashforthTimeStepper
stepper = TwoRateAdamsBashforthTimeStepper(
self.method, dt, self.step_ratio, self.order)
error = self.get_error(stepper, dt, "mrab-%d.dat" % self.order)
eocrec.add_data_point(1/dt, error)
print "------------------------------------------------------"
print "ORDER %d" % self.order
print "------------------------------------------------------"
print eocrec.pretty_print()
orderest = eocrec.estimate_order_of_convergence()[0,1]
assert orderest > self.order*0.70
def main(write_output=True):
from pytools import add_python_path_relative_to_script
add_python_path_relative_to_script("..")
from hedge.backends import guess_run_context
rcon = guess_run_context()
from hedge.tools import EOCRecorder
eoc_rec = EOCRecorder()
if rcon.is_head_rank:
from hedge.mesh.generator import \
make_rect_mesh, \
make_centered_regular_rect_mesh
refine = 4
mesh = make_centered_regular_rect_mesh((0,-5), (10,5), n=(9,9),
post_refine_factor=refine)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [3, 4, 5]:
from gas_dynamics_initials import Vortex
flow = Vortex()
from hedge.models.gas_dynamics import (
GasDynamicsOperator, PolytropeEOS, GammaLawEOS)
from hedge.mesh import TAG_ALL
# works equally well for GammaLawEOS
op = GasDynamicsOperator(dimensions=2, mu=flow.mu,
prandtl=flow.prandtl, spec_gas_const=flow.spec_gas_const,
equation_of_state=PolytropeEOS(flow.gamma),
bc_inflow=flow, bc_outflow=flow, bc_noslip=flow,
inflow_tag=TAG_ALL, source=None)
discr = rcon.make_discretization(mesh_data, order=order,
default_scalar_type=numpy.float64,
quad_min_degrees={
"gasdyn_vol": 3*order,
"gasdyn_face": 3*order,
},
tune_for=op.op_template(),
debug=["cuda_no_plan"])
from hedge.visualization import SiloVisualizer, VtkVisualizer
vis = VtkVisualizer(discr, rcon, "vortex-%d" % order)
#vis = SiloVisualizer(discr, rcon)
fields = flow.volume_interpolant(0, discr)
euler_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
# limiter ------------------------------------------------------------
from hedge.models.gas_dynamics import SlopeLimiter1NEuler
limiter = SlopeLimiter1NEuler(discr, flow.gamma, 2, op)
from hedge.timestep.runge_kutta import SSP3TimeStepper
#stepper = SSP3TimeStepper(limiter=limiter)
stepper = SSP3TimeStepper(
vector_primitive_factory=discr.get_vector_primitive_factory())
#from hedge.timestep import RK4TimeStepper
#stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_file_name = "euler-%d.dat" % order
else:
log_file_name = None
logmgr = LogManager(log_file_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
final_time = flow.final_time
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time, logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(discr,
stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
print "run until t=%g" % final_time
for step, t, dt in step_it:
if step % 10 == 0 and write_output:
#if False:
visf = vis.make_file("vortex-%d-%04d" % (order, step))
#true_fields = vortex.volume_interpolant(t, discr)
from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(visf,
[
("rho", discr.convert_volume(op.rho(fields), kind="numpy")),
("e", discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u", discr.convert_volume(op.rho_u(fields), kind="numpy")),
("u", discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", discr.convert_volume(op.rho(true_fields), kind="numpy")),
#("true_e", discr.convert_volume(op.e(true_fields), kind="numpy")),
#("true_rho_u", discr.convert_volume(op.rho_u(true_fields), kind="numpy")),
#("true_u", discr.convert_volume(op.u(true_fields), kind="numpy")),
#("rhs_rho", discr.convert_volume(op.rho(rhs_fields), kind="numpy")),
#("rhs_e", discr.convert_volume(op.e(rhs_fields), kind="numpy")),
#("rhs_rho_u", discr.convert_volume(op.rho_u(rhs_fields), kind="numpy")),
],
#expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
#("p", "0.4*(e- 0.5*(rho_u*u))"),
#],
time=t, step=step
)
visf.close()
fields = stepper(fields, t, dt, rhs)
#fields = limiter(fields)
assert not numpy.isnan(numpy.sum(fields[0]))
true_fields = flow.volume_interpolant(final_time, discr)
l2_error = discr.norm(fields-true_fields)
l2_error_rho = discr.norm(op.rho(fields)-op.rho(true_fields))
l2_error_e = discr.norm(op.e(fields)-op.e(true_fields))
l2_error_rhou = discr.norm(op.rho_u(fields)-op.rho_u(true_fields))
l2_error_u = discr.norm(op.u(fields)-op.u(true_fields))
eoc_rec.add_data_point(order, l2_error)
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
logmgr.set_constant("l2_error", l2_error)
logmgr.set_constant("l2_error_rho", l2_error_rho)
logmgr.set_constant("l2_error_e", l2_error_e)
logmgr.set_constant("l2_error_rhou", l2_error_rhou)
logmgr.set_constant("l2_error_u", l2_error_u)
logmgr.set_constant("refinement", refine)
finally:
if write_output:
vis.close()
logmgr.close()
discr.close()
# after order loop
assert eoc_rec.estimate_order_of_convergence()[0,1] > 6
def run_convergence_test_advec(dtype,
flux_type,
random_partition,
mesh_gen,
debug_output=False):
"""Test whether 2/3D advection actually converges"""
from hedge.timestep import RK4TimeStepper
from hedge.tools import EOCRecorder
from math import sin
from hedge.data import TimeDependentGivenFunction
from hedge.visualization import SiloVisualizer
from hedge.backends import guess_run_context
rcon = guess_run_context(["mpi"])
# note: x component must remain zero because x-periodicity is used
v = np.array([0.0, 0.9, 0.3])
def f(x):
return sin(x)
def u_analytic(x, el, t):
return f(
(np.dot(-v[:dims], x) / la.norm(v[:dims]) + t * la.norm(v[:dims])))
def boundary_tagger(vertices, el, face_nr, points):
face_normal = el.face_normals[face_nr]
if np.dot(face_normal, v[:len(face_normal)]) < 0:
return ["inflow"]
else:
return ["outflow"]
mesh = mesh_gen(boundary_tagger)
eoc_rec = EOCRecorder()
if random_partition:
# Distribute elements randomly across nodes.
# This is bad, efficiency-wise, but it puts stress
# on the parallel implementation, which is desired here.
# Another main point of this is to force the code to split
# a periodic face pair across nodes.
from random import choice
partition = [choice(rcon.ranks) for el in mesh.elements]
else:
partition = None
for order in [1, 2, 3, 4]:
if rcon.is_head_rank:
mesh_data = rcon.distribute_mesh(mesh, partition)
else:
mesh_data = rcon.receive_mesh()
dims = mesh.points.shape[1]
discr = rcon.make_discretization(mesh_data,
order=order,
default_scalar_type=dtype)
op = StrongAdvectionOperator(
v[:dims],
inflow_u=TimeDependentGivenFunction(u_analytic),
flux_type=flux_type)
if debug_output:
vis = SiloVisualizer(discr, rcon)
u = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, 0))
ic = u.copy()
if debug_output and rcon.is_head_rank:
print "#elements=%d" % len(mesh.elements)
test_name = "test-%s-o%d-m%s-r%s" % (
flux_type, order, mesh_gen.__name__, random_partition)
rhs = op.bind(discr)
stepper = RK4TimeStepper(dtype=dtype)
from hedge.timestep import times_and_steps
final_time = 1
step_it = times_and_steps(final_time=final_time,
max_dt_getter=lambda t: op.estimate_timestep(
discr, stepper=stepper, t=t, fields=u))
for step, t, dt in step_it:
u = stepper(u, t, dt, rhs)
assert u.dtype == dtype
u_true = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, final_time))
error = u - u_true
l2_error = discr.norm(error)
if debug_output:
visf = vis.make_file(test_name + "-final")
vis.add_data(visf, [("u", u), ("u_true", u_true), ("ic", ic)])
visf.close()
eoc_rec.add_data_point(order, l2_error)
if debug_output and rcon.is_head_rank:
print "%s\n%s\n" % (flux_type.upper(), "-" * len(flux_type))
print eoc_rec.pretty_print(abscissa_label="Poly. Order",
error_label="L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 3
assert eoc_rec.estimate_order_of_convergence(2)[-1, 1] > 7
def main(write_output=True, allow_features=None, flux_type_arg=1, bdry_flux_type_arg=None, extra_discr_args={}):
from hedge.mesh.generator import make_cylinder_mesh, make_box_mesh
from hedge.tools import EOCRecorder, to_obj_array
from math import sqrt, pi
from analytic_solutions import (
check_time_harmonic_solution,
RealPartAdapter,
SplitComplexAdapter,
CylindricalFieldAdapter,
CylindricalCavityMode,
RectangularWaveguideMode,
RectangularCavityMode,
)
from hedge.models.em import MaxwellOperator
from hedge.backends import guess_run_context
rcon = guess_run_context(allow_features)
epsilon0 = 8.8541878176e-12 # C**2 / (N m**2)
mu0 = 4 * pi * 1e-7 # N/A**2.
epsilon = 1 * epsilon0
mu = 1 * mu0
eoc_rec = EOCRecorder()
cylindrical = False
periodic = False
if cylindrical:
R = 1
d = 2
mode = CylindricalCavityMode(m=1, n=1, p=1, radius=R, height=d, epsilon=epsilon, mu=mu)
r_sol = CylindricalFieldAdapter(RealPartAdapter(mode))
c_sol = SplitComplexAdapter(CylindricalFieldAdapter(mode))
if rcon.is_head_rank:
mesh = make_cylinder_mesh(radius=R, height=d, max_volume=0.01)
else:
if periodic:
mode = RectangularWaveguideMode(epsilon, mu, (3, 2, 1))
periodicity = (False, False, True)
else:
periodicity = None
mode = RectangularCavityMode(epsilon, mu, (1, 2, 2))
if rcon.is_head_rank:
mesh = make_box_mesh(max_volume=0.001, periodicity=periodicity)
if rcon.is_head_rank:
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [4, 5, 6]:
# for order in [1,2,3,4,5,6]:
extra_discr_args.setdefault("debug", []).extend(["cuda_no_plan", "cuda_dump_kernels"])
op = MaxwellOperator(epsilon, mu, flux_type=flux_type_arg, bdry_flux_type=bdry_flux_type_arg)
discr = rcon.make_discretization(mesh_data, order=order, tune_for=op.op_template(), **extra_discr_args)
from hedge.visualization import VtkVisualizer
if write_output:
vis = VtkVisualizer(discr, rcon, "em-%d" % order)
mode.set_time(0)
def get_true_field():
return discr.convert_volume(
to_obj_array(mode(discr).real.astype(discr.default_scalar_type).copy()), kind=discr.compute_kind
)
fields = get_true_field()
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
from hedge.timestep.runge_kutta import LSRK4TimeStepper
stepper = LSRK4TimeStepper(dtype=discr.default_scalar_type, rcon=rcon)
# from hedge.timestep.dumka3 import Dumka3TimeStepper
# stepper = Dumka3TimeStepper(3, dtype=discr.default_scalar_type, rcon=rcon)
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, add_simulation_quantities, add_run_info
if write_output:
log_file_name = "maxwell-%d.dat" % order
else:
log_file_name = None
logmgr = LogManager(log_file_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
from pytools.log import IntervalTimer
vis_timer = IntervalTimer("t_vis", "Time spent visualizing")
logmgr.add_quantity(vis_timer)
from hedge.log import EMFieldGetter, add_em_quantities
field_getter = EMFieldGetter(discr, op, lambda: fields)
add_em_quantities(logmgr, op, field_getter)
logmgr.add_watches(["step.max", "t_sim.max", ("W_field", "W_el+W_mag"), "t_step.max"])
# timestep loop -------------------------------------------------------
rhs = op.bind(discr)
final_time = 0.5e-9
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time,
logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(discr, stepper=stepper, t=t, fields=fields),
)
for step, t, dt in step_it:
if step % 50 == 0 and write_output:
sub_timer = vis_timer.start_sub_timer()
e, h = op.split_eh(fields)
visf = vis.make_file("em-%d-%04d" % (order, step))
vis.add_data(
visf,
[("e", discr.convert_volume(e, kind="numpy")), ("h", discr.convert_volume(h, kind="numpy"))],
time=t,
step=step,
)
visf.close()
sub_timer.stop().submit()
fields = stepper(fields, t, dt, rhs)
mode.set_time(final_time)
eoc_rec.add_data_point(order, discr.norm(fields - get_true_field()))
finally:
if write_output:
vis.close()
logmgr.close()
discr.close()
if rcon.is_head_rank:
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 6
def main():
from hedge.backends import guess_run_context
rcon = guess_run_context(
#["cuda"]
)
from hedge.tools import EOCRecorder, to_obj_array
eoc_rec = EOCRecorder()
def boundary_tagger(vertices, el, face_nr, all_v):
return ["inflow"]
if rcon.is_head_rank:
from hedge.mesh import make_rect_mesh, \
make_centered_regular_rect_mesh
#mesh = make_rect_mesh((0,0), (10,1), max_area=0.01)
refine = 1
mesh = make_centered_regular_rect_mesh(
(0, 0),
(10, 1),
n=(20, 4),
#periodicity=(True, False),
post_refine_factor=refine,
boundary_tagger=boundary_tagger)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [3]:
discr = rcon.make_discretization(mesh_data,
order=order,
default_scalar_type=numpy.float64)
from hedge.visualization import SiloVisualizer, VtkVisualizer
#vis = VtkVisualizer(discr, rcon, "shearflow-%d" % order)
vis = SiloVisualizer(discr, rcon)
shearflow = SteadyShearFlow()
fields = shearflow.volume_interpolant(0, discr)
gamma, mu, prandtl, spec_gas_const = shearflow.properties()
from hedge.models.gas_dynamics import GasDynamicsOperator
op = GasDynamicsOperator(dimensions=2,
gamma=gamma,
mu=mu,
prandtl=prandtl,
spec_gas_const=spec_gas_const,
bc_inflow=shearflow,
bc_outflow=shearflow,
bc_noslip=shearflow,
inflow_tag="inflow",
outflow_tag="outflow",
noslip_tag="noslip")
navierstokes_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = navierstokes_ex(t, q)
max_eigval[0] = speed
return ode_rhs
# needed to get first estimate of maximum eigenvalue
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
from hedge.timestep import RK4TimeStepper
stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
logmgr = LogManager("navierstokes-cpu-%d-%d.dat" % (order, refine),
"w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=0.3,
#max_steps=500,
logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(
discr, stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
for step, t, dt in step_it:
if step % 10 == 0:
#if False:
visf = vis.make_file("shearflow-%d-%04d" % (order, step))
#true_fields = shearflow.volume_interpolant(t, discr)
from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(
visf,
[
("rho",
discr.convert_volume(op.rho(fields),
kind="numpy")),
("e",
discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u",
discr.convert_volume(op.rho_u(fields),
kind="numpy")),
("u",
discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", discr.convert_volume(op.rho(true_fields), kind="numpy")),
#("true_e", discr.convert_volume(op.e(true_fields), kind="numpy")),
#("true_rho_u", discr.convert_volume(op.rho_u(true_fields), kind="numpy")),
#("true_u", discr.convert_volume(op.u(true_fields), kind="numpy")),
],
expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
("p", "0.4*(e- 0.5*(rho_u*u))"),
],
time=t,
step=step)
visf.close()
fields = stepper(fields, t, dt, rhs)
true_fields = shearflow.volume_interpolant(t, discr)
l2_error = discr.norm(op.u(fields) - op.u(true_fields))
eoc_rec.add_data_point(order, l2_error)
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
logmgr.set_constant("l2_error", l2_error)
finally:
vis.close()
logmgr.save()
discr.close()
def main(write_output=True):
from pytools import add_python_path_relative_to_script
add_python_path_relative_to_script("..")
from hedge.backends import guess_run_context
rcon = guess_run_context()
from hedge.tools import EOCRecorder
eoc_rec = EOCRecorder()
if rcon.is_head_rank:
from hedge.mesh.generator import \
make_rect_mesh, \
make_centered_regular_rect_mesh
refine = 4
mesh = make_centered_regular_rect_mesh((0, -5), (10, 5),
n=(9, 9),
post_refine_factor=refine)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [3, 4, 5]:
from gas_dynamics_initials import Vortex
flow = Vortex()
from hedge.models.gas_dynamics import (GasDynamicsOperator,
PolytropeEOS, GammaLawEOS)
from hedge.mesh import TAG_ALL
# works equally well for GammaLawEOS
op = GasDynamicsOperator(dimensions=2,
mu=flow.mu,
prandtl=flow.prandtl,
spec_gas_const=flow.spec_gas_const,
equation_of_state=PolytropeEOS(flow.gamma),
bc_inflow=flow,
bc_outflow=flow,
bc_noslip=flow,
inflow_tag=TAG_ALL,
source=None)
discr = rcon.make_discretization(mesh_data,
order=order,
default_scalar_type=numpy.float64,
quad_min_degrees={
"gasdyn_vol": 3 * order,
"gasdyn_face": 3 * order,
},
tune_for=op.op_template(),
debug=["cuda_no_plan"])
from hedge.visualization import SiloVisualizer, VtkVisualizer
vis = VtkVisualizer(discr, rcon, "vortex-%d" % order)
#vis = SiloVisualizer(discr, rcon)
fields = flow.volume_interpolant(0, discr)
euler_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
# limiter ------------------------------------------------------------
from hedge.models.gas_dynamics import SlopeLimiter1NEuler
limiter = SlopeLimiter1NEuler(discr, flow.gamma, 2, op)
from hedge.timestep.runge_kutta import SSP3TimeStepper
#stepper = SSP3TimeStepper(limiter=limiter)
stepper = SSP3TimeStepper(
vector_primitive_factory=discr.get_vector_primitive_factory())
#from hedge.timestep import RK4TimeStepper
#stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_file_name = "euler-%d.dat" % order
else:
log_file_name = None
logmgr = LogManager(log_file_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
final_time = flow.final_time
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time,
logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(
discr, stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
print "run until t=%g" % final_time
for step, t, dt in step_it:
if step % 10 == 0 and write_output:
#if False:
visf = vis.make_file("vortex-%d-%04d" % (order, step))
#true_fields = vortex.volume_interpolant(t, discr)
from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(
visf,
[
("rho",
discr.convert_volume(op.rho(fields),
kind="numpy")),
("e",
discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u",
discr.convert_volume(op.rho_u(fields),
kind="numpy")),
("u",
discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", discr.convert_volume(op.rho(true_fields), kind="numpy")),
#("true_e", discr.convert_volume(op.e(true_fields), kind="numpy")),
#("true_rho_u", discr.convert_volume(op.rho_u(true_fields), kind="numpy")),
#("true_u", discr.convert_volume(op.u(true_fields), kind="numpy")),
#("rhs_rho", discr.convert_volume(op.rho(rhs_fields), kind="numpy")),
#("rhs_e", discr.convert_volume(op.e(rhs_fields), kind="numpy")),
#("rhs_rho_u", discr.convert_volume(op.rho_u(rhs_fields), kind="numpy")),
],
#expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
#("p", "0.4*(e- 0.5*(rho_u*u))"),
#],
time=t,
step=step)
visf.close()
fields = stepper(fields, t, dt, rhs)
#fields = limiter(fields)
assert not numpy.isnan(numpy.sum(fields[0]))
true_fields = flow.volume_interpolant(final_time, discr)
l2_error = discr.norm(fields - true_fields)
l2_error_rho = discr.norm(op.rho(fields) - op.rho(true_fields))
l2_error_e = discr.norm(op.e(fields) - op.e(true_fields))
l2_error_rhou = discr.norm(
op.rho_u(fields) - op.rho_u(true_fields))
l2_error_u = discr.norm(op.u(fields) - op.u(true_fields))
eoc_rec.add_data_point(order, l2_error)
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
logmgr.set_constant("l2_error", l2_error)
logmgr.set_constant("l2_error_rho", l2_error_rho)
logmgr.set_constant("l2_error_e", l2_error_e)
logmgr.set_constant("l2_error_rhou", l2_error_rhou)
logmgr.set_constant("l2_error_u", l2_error_u)
logmgr.set_constant("refinement", refine)
finally:
if write_output:
vis.close()
logmgr.close()
discr.close()
# after order loop
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 6
def test_convergence_advec_2d():
"""Test whether 2D advection actually converges"""
import pyublas # noqa
from hedge.mesh.generator import make_disk_mesh, make_regular_rect_mesh
from hedge.discretization.local import TriangleDiscretization
from hedge.timestep import RK4TimeStepper
from hedge.tools import EOCRecorder
from math import sin, pi
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
v = numpy.array([0.27, 0])
norm_a = la.norm(v)
from numpy import dot
def f(x):
return sin(x)
def u_analytic(x, el, t):
return f((-dot(v, x) / norm_a + t * norm_a))
def boundary_tagger(vertices, el, face_nr, all_v):
if dot(el.face_normals[face_nr], v) < 0:
return ["inflow"]
else:
return ["outflow"]
for mesh in [
# non-periodic
make_disk_mesh(r=pi, boundary_tagger=boundary_tagger,
max_area=0.5),
# periodic
make_regular_rect_mesh(
a=(0, 0),
b=(2 * pi, 1),
n=(8, 4),
periodicity=(True, False),
boundary_tagger=boundary_tagger,
)
]:
for flux_type in StrongAdvectionOperator.flux_types:
eoc_rec = EOCRecorder()
for order in [1, 2, 3, 4, 5, 6]:
discr = discr_class(
mesh,
TriangleDiscretization(order),
debug=discr_class.noninteractive_debug_flags())
op = StrongAdvectionOperator(
v,
inflow_u=TimeDependentGivenFunction(u_analytic),
flux_type=flux_type)
u = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, 0))
stepper = RK4TimeStepper()
dt = op.estimate_timestep(discr, stepper=stepper)
nsteps = int(1 / dt)
rhs = op.bind(discr)
for step in range(nsteps):
u = stepper(u, step * dt, dt, rhs)
u_true = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, nsteps * dt))
error = u - u_true
error_l2 = discr.norm(error)
eoc_rec.add_data_point(order, error_l2)
if False:
print "%s\n%s\n" % (flux_type.upper(), "-" * len(flux_type))
print eoc_rec.pretty_print(abscissa_label="Poly. Order",
error_label="L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 4
assert eoc_rec.estimate_order_of_convergence(2)[-1, 1] > 10
def main(final_time=1, write_output=False):
from hedge.backends import guess_run_context
rcon = guess_run_context()
from hedge.tools import EOCRecorder, to_obj_array
eoc_rec = EOCRecorder()
if rcon.is_head_rank:
from hedge.mesh import make_box_mesh
mesh = make_box_mesh((0,0,0), (10,10,10), max_volume=0.5)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [3, 4, 5]:
discr = rcon.make_discretization(mesh_data, order=order,
default_scalar_type=numpy.float64)
from hedge.visualization import SiloVisualizer, VtkVisualizer
vis = VtkVisualizer(discr, rcon, "sinewave-%d" % order)
#vis = SiloVisualizer(discr, rcon)
sinewave = SineWave()
fields = sinewave.volume_interpolant(0, discr)
gamma, mu, prandtl, spec_gas_const = sinewave.properties()
from hedge.mesh import TAG_ALL
from hedge.models.gas_dynamics import GasDynamicsOperator
op = GasDynamicsOperator(dimensions=mesh.dimensions, gamma=gamma, mu=mu,
prandtl=prandtl, spec_gas_const=spec_gas_const,
bc_inflow=sinewave, bc_outflow=sinewave, bc_noslip=sinewave,
inflow_tag=TAG_ALL, source=None)
euler_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
from hedge.timestep import RK4TimeStepper
stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_name = ("euler-sinewave-%(order)d-%(els)d.dat"
% {"order":order, "els":len(mesh.elements)})
else:
log_name = False
logmgr = LogManager(log_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time, logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(discr,
stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
for step, t, dt in step_it:
#if step % 10 == 0:
if write_output:
visf = vis.make_file("sinewave-%d-%04d" % (order, step))
#from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(visf,
[
("rho", discr.convert_volume(op.rho(fields), kind="numpy")),
("e", discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u", discr.convert_volume(op.rho_u(fields), kind="numpy")),
("u", discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", op.rho(true_fields)),
#("true_e", op.e(true_fields)),
#("true_rho_u", op.rho_u(true_fields)),
#("true_u", op.u(true_fields)),
#("rhs_rho", op.rho(rhs_fields)),
#("rhs_e", op.e(rhs_fields)),
#("rhs_rho_u", op.rho_u(rhs_fields)),
],
#expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
#("p", "0.4*(e- 0.5*(rho_u*u))"),
#],
time=t, step=step
)
visf.close()
fields = stepper(fields, t, dt, rhs)
finally:
vis.close()
logmgr.close()
discr.close()
true_fields = sinewave.volume_interpolant(t, discr)
eoc_rec.add_data_point(order, discr.norm(fields-true_fields))
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
def main(write_output=True):
from pytools import add_python_path_relative_to_script
add_python_path_relative_to_script("..")
from hedge.backends import guess_run_context
rcon = guess_run_context()
from hedge.tools import EOCRecorder
eoc_rec = EOCRecorder()
if rcon.is_head_rank:
from hedge.mesh.generator import \
make_rect_mesh, \
make_centered_regular_rect_mesh
refine = 4
mesh = make_centered_regular_rect_mesh((0,-5), (10,5), n=(9,9),
post_refine_factor=refine)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
# a second mesh to regrid to
if rcon.is_head_rank:
from hedge.mesh.generator import \
make_rect_mesh, \
make_centered_regular_rect_mesh
refine = 4
mesh2 = make_centered_regular_rect_mesh((0,-5), (10,5), n=(8,8),
post_refine_factor=refine)
mesh_data2 = rcon.distribute_mesh(mesh2)
else:
mesh_data2 = rcon.receive_mesh()
for order in [3,4]:
discr = rcon.make_discretization(mesh_data, order=order,
default_scalar_type=numpy.float64,
quad_min_degrees={
"gasdyn_vol": 3*order,
"gasdyn_face": 3*order,
})
discr2 = rcon.make_discretization(mesh_data2, order=order,
default_scalar_type=numpy.float64,
quad_min_degrees={
"gasdyn_vol": 3*order,
"gasdyn_face": 3*order,
})
from hedge.visualization import SiloVisualizer, VtkVisualizer
vis = VtkVisualizer(discr, rcon, "vortex-%d" % order)
#vis = SiloVisualizer(discr, rcon)
from gas_dynamics_initials import Vortex
vortex = Vortex()
fields = vortex.volume_interpolant(0, discr)
from hedge.models.gas_dynamics import GasDynamicsOperator
from hedge.mesh import TAG_ALL
op = GasDynamicsOperator(dimensions=2, gamma=vortex.gamma, mu=vortex.mu,
prandtl=vortex.prandtl, spec_gas_const=vortex.spec_gas_const,
bc_inflow=vortex, bc_outflow=vortex, bc_noslip=vortex,
inflow_tag=TAG_ALL, source=None)
euler_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements for mesh 1 =", len(mesh.elements)
print "#elements for mesh 2 =", len(mesh2.elements)
# limiter ------------------------------------------------------------
from hedge.models.gas_dynamics import SlopeLimiter1NEuler
limiter = SlopeLimiter1NEuler(discr, vortex.gamma, 2, op)
from hedge.timestep import SSPRK3TimeStepper
#stepper = SSPRK3TimeStepper(limiter=limiter)
stepper = SSPRK3TimeStepper()
#from hedge.timestep import RK4TimeStepper
#stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_file_name = "euler-%d.dat" % order
else:
log_file_name = None
logmgr = LogManager(log_file_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
final_time = 0.2
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time, logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(discr,
stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
for step, t, dt in step_it:
if step % 10 == 0 and write_output:
#if False:
visf = vis.make_file("vortex-%d-%04d" % (order, step))
#true_fields = vortex.volume_interpolant(t, discr)
from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(visf,
[
("rho", discr.convert_volume(op.rho(fields), kind="numpy")),
("e", discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u", discr.convert_volume(op.rho_u(fields), kind="numpy")),
("u", discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", discr.convert_volume(op.rho(true_fields), kind="numpy")),
#("true_e", discr.convert_volume(op.e(true_fields), kind="numpy")),
#("true_rho_u", discr.convert_volume(op.rho_u(true_fields), kind="numpy")),
#("true_u", discr.convert_volume(op.u(true_fields), kind="numpy")),
#("rhs_rho", discr.convert_volume(op.rho(rhs_fields), kind="numpy")),
#("rhs_e", discr.convert_volume(op.e(rhs_fields), kind="numpy")),
#("rhs_rho_u", discr.convert_volume(op.rho_u(rhs_fields), kind="numpy")),
],
#expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
#("p", "0.4*(e- 0.5*(rho_u*u))"),
#],
time=t, step=step
)
visf.close()
fields = stepper(fields, t, dt, rhs)
#fields = limiter(fields)
#regrid to discr2 at some arbitrary time
if step == 21:
#get interpolated fields
fields = discr.get_regrid_values(fields, discr2, dtype=None, use_btree=True, thresh=1e-8)
#get new stepper (old one has reference to discr
stepper = SSPRK3TimeStepper()
#new bind
euler_ex = op.bind(discr2)
#new rhs
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(t+dt, fields)
#add logmanager
#discr2.add_instrumentation(logmgr)
#new step_it
step_it = times_and_steps(
final_time=final_time, logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(discr2,
stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
#new visualization
vis.close()
vis = VtkVisualizer(discr2, rcon, "vortexNewGrid-%d" % order)
discr=discr2
assert not numpy.isnan(numpy.sum(fields[0]))
true_fields = vortex.volume_interpolant(final_time, discr)
l2_error = discr.norm(fields-true_fields)
l2_error_rho = discr.norm(op.rho(fields)-op.rho(true_fields))
l2_error_e = discr.norm(op.e(fields)-op.e(true_fields))
l2_error_rhou = discr.norm(op.rho_u(fields)-op.rho_u(true_fields))
l2_error_u = discr.norm(op.u(fields)-op.u(true_fields))
eoc_rec.add_data_point(order, l2_error)
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
logmgr.set_constant("l2_error", l2_error)
logmgr.set_constant("l2_error_rho", l2_error_rho)
logmgr.set_constant("l2_error_e", l2_error_e)
logmgr.set_constant("l2_error_rhou", l2_error_rhou)
logmgr.set_constant("l2_error_u", l2_error_u)
logmgr.set_constant("refinement", refine)
finally:
if write_output:
vis.close()
logmgr.close()
discr.close()
def run_convergence_test_advec(dtype, debug_output=False):
"""Test whether 2/3D advection actually converges"""
from hedge.mesh.generator import make_ball_mesh, make_box_mesh, make_rect_mesh
from hedge.timestep import RK4TimeStepper
from hedge.tools import EOCRecorder
from math import sin, pi, sqrt
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
from hedge.visualization import SiloVisualizer
from hedge.backends import guess_run_context
rcon = guess_run_context(["mpi"])
# note: x component must remain zero because x-periodicity is used
v = numpy.array([0.0,0.9,0.3])
def f(x):
return sin(x)
def u_analytic(x, el, t):
return f((numpy.dot(-v[:dims],x)/la.norm(v[:dims])+t*la.norm(v[:dims])))
def boundary_tagger(vertices, el, face_nr, points):
face_normal = el.face_normals[face_nr]
if numpy.dot(face_normal, v[:len(face_normal)]) < 0:
return ["inflow"]
else:
return ["outflow"]
for i_mesh, mesh in enumerate([
# 2D semiperiodic
make_rect_mesh(b=(2*pi,3), max_area=0.4,
periodicity=(True, False),
subdivisions=(5,10),
boundary_tagger=boundary_tagger,
),
# 3D x-periodic
make_box_mesh((0,0,0), (2*pi, 2, 2), max_volume=0.4,
periodicity=(True, False, False),
boundary_tagger=boundary_tagger,
),
# non-periodic
make_ball_mesh(r=pi,
boundary_tagger=boundary_tagger, max_volume=0.7),
]):
for flux_type in StrongAdvectionOperator.flux_types:
for random_partition in [True, False]:
eoc_rec = EOCRecorder()
if random_partition:
# Distribute elements randomly across nodes.
# This is bad, efficiency-wise, but it puts stress
# on the parallel implementation, which is desired here.
# Another main point of this is to force the code to split
# a periodic face pair across nodes.
from random import choice
partition = [choice(rcon.ranks) for el in mesh.elements]
else:
partition = None
for order in [1,2,3,4]:
if rcon.is_head_rank:
mesh_data = rcon.distribute_mesh(mesh, partition)
else:
mesh_data = rcon.receive_mesh()
dims = mesh.points.shape[1]
discr = rcon.make_discretization(mesh_data, order=order,
default_scalar_type=dtype)
op = StrongAdvectionOperator(v[:dims],
inflow_u=TimeDependentGivenFunction(u_analytic),
flux_type=flux_type)
if debug_output:
vis = SiloVisualizer(discr, rcon)
u = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, 0))
ic = u.copy()
if debug_output and rcon.is_head_rank:
print "#elements=%d" % len(mesh.elements)
test_name = "test-%s-o%d-m%d-r%s" % (
flux_type, order, i_mesh, random_partition)
rhs = op.bind(discr)
stepper = RK4TimeStepper(dtype=dtype)
from hedge.timestep import times_and_steps
final_time = 1
step_it = times_and_steps(
final_time=final_time,
max_dt_getter=lambda t: op.estimate_timestep(discr,
stepper=stepper, t=t, fields=u))
for step, t, dt in step_it:
u = stepper(u, t, dt, rhs)
assert u.dtype == dtype
u_true = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, final_time))
error = u-u_true
l2_error = discr.norm(error)
if debug_output:
visf = vis.make_file(test_name+"-final")
vis.add_data(visf, [
("u", u),
("u_true", u_true),
("ic", ic)])
visf.close()
eoc_rec.add_data_point(order, l2_error)
if debug_output and rcon.is_head_rank:
print "%s\n%s\n" % (flux_type.upper(), "-" * len(flux_type))
print eoc_rec.pretty_print(abscissa_label="Poly. Order",
error_label="L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0,1] > 3
assert eoc_rec.estimate_order_of_convergence(2)[-1,1] > 7
def test_elliptic():
"""Test various properties of elliptic operators."""
from hedge.tools import unit_vector
def matrix_rep(op):
h, w = op.shape
mat = numpy.zeros(op.shape)
for j in range(w):
mat[:, j] = op(unit_vector(w, j))
return mat
def check_grad_mat():
import pyublas
if not pyublas.has_sparse_wrappers():
return
grad_mat = op.grad_matrix()
# print len(discr), grad_mat.nnz, type(grad_mat)
for i in range(10):
u = numpy.random.randn(len(discr))
mat_result = grad_mat * u
op_result = numpy.hstack(op.grad(u))
err = la.norm(mat_result - op_result) * la.norm(op_result)
assert la.norm(mat_result - op_result) * la.norm(op_result) < 1e-5
def check_matrix_tgt():
big = num.zeros((20, 20), flavor=num.SparseBuildMatrix)
small = num.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print small
from hedge._internal import MatrixTarget
tgt = MatrixTarget(big, 4, 4)
tgt.begin(small.shape[0], small.shape[1])
print "YO"
tgt.add_coefficients(4, 4, small)
print "DUDE"
tgt.finalize()
print big
import pymbolic
v_x = pymbolic.var("x")
truesol = pymbolic.parse("math.sin(x[0]**2*x[1]**2)")
truesol_c = pymbolic.compile(truesol, variables=["x"])
rhs = pymbolic.simplify(pymbolic.laplace(truesol, [v_x[0], v_x[1]]))
rhs_c = pymbolic.compile(rhs, variables=["x", "el"])
from hedge.mesh import TAG_ALL, TAG_NONE
from hedge.mesh.generator import make_disk_mesh
mesh = make_disk_mesh(r=0.5, max_area=0.1, faces=20)
mesh = mesh.reordered_by("cuthill")
from hedge.backends import CPURunContext
rcon = CPURunContext()
from hedge.tools import EOCRecorder
eocrec = EOCRecorder()
for order in [1, 2, 3, 4, 5]:
for flux in ["ldg", "ip"]:
from hedge.discretization.local import TriangleDiscretization
discr = rcon.make_discretization(
mesh, TriangleDiscretization(order), debug=discr_class.noninteractive_debug_flags()
)
from hedge.data import GivenFunction
from hedge.models.poisson import PoissonOperator
op = PoissonOperator(
discr.dimensions,
dirichlet_tag=TAG_ALL,
dirichlet_bc=GivenFunction(lambda x, el: truesol_c(x)),
neumann_tag=TAG_NONE,
)
bound_op = op.bind(discr)
if order <= 3:
mat = matrix_rep(bound_op)
sym_err = la.norm(mat - mat.T)
# print sym_err
assert sym_err < 1e-12
# check_grad_mat()
from hedge.iterative import parallel_cg
truesol_v = discr.interpolate_volume_function(lambda x, el: truesol_c(x))
sol_v = -parallel_cg(
rcon,
-bound_op,
bound_op.prepare_rhs(discr.interpolate_volume_function(rhs_c)),
tol=1e-10,
max_iterations=40000,
)
eocrec.add_data_point(order, discr.norm(sol_v - truesol_v))
# print eocrec.pretty_print()
assert eocrec.estimate_order_of_convergence()[0, 1] > 8
def test_convergence_advec_2d():
"""Test whether 2D advection actually converges"""
import pyublas
from hedge.mesh.generator import make_disk_mesh, make_regular_rect_mesh
from hedge.discretization.local import TriangleDiscretization
from hedge.timestep import RK4TimeStepper
from hedge.tools import EOCRecorder
from math import sin, pi, sqrt
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
v = numpy.array([0.27, 0])
norm_a = la.norm(v)
from numpy import dot
def f(x):
return sin(x)
def u_analytic(x, el, t):
return f((-dot(v, x) / norm_a + t * norm_a))
def boundary_tagger(vertices, el, face_nr, all_v):
if dot(el.face_normals[face_nr], v) < 0:
return ["inflow"]
else:
return ["outflow"]
for mesh in [
# non-periodic
make_disk_mesh(r=pi, boundary_tagger=boundary_tagger, max_area=0.5),
# periodic
make_regular_rect_mesh(
a=(0, 0), b=(2 * pi, 1), n=(8, 4), periodicity=(True, False), boundary_tagger=boundary_tagger
),
]:
for flux_type in StrongAdvectionOperator.flux_types:
eoc_rec = EOCRecorder()
for order in [1, 2, 3, 4, 5, 6]:
discr = discr_class(mesh, TriangleDiscretization(order), debug=discr_class.noninteractive_debug_flags())
op = StrongAdvectionOperator(v, inflow_u=TimeDependentGivenFunction(u_analytic), flux_type=flux_type)
u = discr.interpolate_volume_function(lambda x, el: u_analytic(x, el, 0))
stepper = RK4TimeStepper()
dt = op.estimate_timestep(discr, stepper=stepper)
nsteps = int(1 / dt)
rhs = op.bind(discr)
for step in range(nsteps):
u = stepper(u, step * dt, dt, rhs)
u_true = discr.interpolate_volume_function(lambda x, el: u_analytic(x, el, nsteps * dt))
error = u - u_true
error_l2 = discr.norm(error)
eoc_rec.add_data_point(order, error_l2)
if False:
print "%s\n%s\n" % (flux_type.upper(), "-" * len(flux_type))
print eoc_rec.pretty_print(abscissa_label="Poly. Order", error_label="L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 4
assert eoc_rec.estimate_order_of_convergence(2)[-1, 1] > 10
def main():
from hedge.backends import guess_run_context
rcon = guess_run_context(
#["cuda"]
)
from hedge.tools import EOCRecorder, to_obj_array
eoc_rec = EOCRecorder()
def boundary_tagger(vertices, el, face_nr, all_v):
return ["inflow"]
if rcon.is_head_rank:
from hedge.mesh import make_rect_mesh, \
make_centered_regular_rect_mesh
#mesh = make_rect_mesh((0,0), (10,1), max_area=0.01)
refine = 1
mesh = make_centered_regular_rect_mesh((0,0), (10,1), n=(20,4),
#periodicity=(True, False),
post_refine_factor=refine,
boundary_tagger=boundary_tagger)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [3]:
discr = rcon.make_discretization(mesh_data, order=order,
default_scalar_type=numpy.float64)
from hedge.visualization import SiloVisualizer, VtkVisualizer
#vis = VtkVisualizer(discr, rcon, "shearflow-%d" % order)
vis = SiloVisualizer(discr, rcon)
shearflow = SteadyShearFlow()
fields = shearflow.volume_interpolant(0, discr)
gamma, mu, prandtl, spec_gas_const = shearflow.properties()
from hedge.models.gas_dynamics import GasDynamicsOperator
op = GasDynamicsOperator(dimensions=2, gamma=gamma, mu=mu,
prandtl=prandtl, spec_gas_const=spec_gas_const,
bc_inflow=shearflow, bc_outflow=shearflow, bc_noslip=shearflow,
inflow_tag="inflow", outflow_tag="outflow", noslip_tag="noslip")
navierstokes_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = navierstokes_ex(t, q)
max_eigval[0] = speed
return ode_rhs
# needed to get first estimate of maximum eigenvalue
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
from hedge.timestep import RK4TimeStepper
stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
logmgr = LogManager("navierstokes-cpu-%d-%d.dat" % (order, refine),
"w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=0.3,
#max_steps=500,
logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(discr,
stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
for step, t, dt in step_it:
if step % 10 == 0:
#if False:
visf = vis.make_file("shearflow-%d-%04d" % (order, step))
#true_fields = shearflow.volume_interpolant(t, discr)
from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(visf,
[
("rho", discr.convert_volume(op.rho(fields), kind="numpy")),
("e", discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u", discr.convert_volume(op.rho_u(fields), kind="numpy")),
("u", discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", discr.convert_volume(op.rho(true_fields), kind="numpy")),
#("true_e", discr.convert_volume(op.e(true_fields), kind="numpy")),
#("true_rho_u", discr.convert_volume(op.rho_u(true_fields), kind="numpy")),
#("true_u", discr.convert_volume(op.u(true_fields), kind="numpy")),
],
expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
("p", "0.4*(e- 0.5*(rho_u*u))"),
],
time=t, step=step
)
visf.close()
fields = stepper(fields, t, dt, rhs)
true_fields = shearflow.volume_interpolant(t, discr)
l2_error = discr.norm(op.u(fields)-op.u(true_fields))
eoc_rec.add_data_point(order, l2_error)
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
logmgr.set_constant("l2_error", l2_error)
finally:
vis.close()
logmgr.save()
discr.close()
def main(final_time=1, write_output=False):
from hedge.backends import guess_run_context
rcon = guess_run_context()
from hedge.tools import EOCRecorder, to_obj_array
eoc_rec = EOCRecorder()
if rcon.is_head_rank:
from hedge.mesh import make_box_mesh
mesh = make_box_mesh((0, 0, 0), (10, 10, 10), max_volume=0.5)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [3, 4, 5]:
discr = rcon.make_discretization(mesh_data,
order=order,
default_scalar_type=numpy.float64)
from hedge.visualization import SiloVisualizer, VtkVisualizer
vis = VtkVisualizer(discr, rcon, "sinewave-%d" % order)
#vis = SiloVisualizer(discr, rcon)
sinewave = SineWave()
fields = sinewave.volume_interpolant(0, discr)
gamma, mu, prandtl, spec_gas_const = sinewave.properties()
from hedge.mesh import TAG_ALL
from hedge.models.gas_dynamics import GasDynamicsOperator
op = GasDynamicsOperator(dimensions=mesh.dimensions,
gamma=gamma,
mu=mu,
prandtl=prandtl,
spec_gas_const=spec_gas_const,
bc_inflow=sinewave,
bc_outflow=sinewave,
bc_noslip=sinewave,
inflow_tag=TAG_ALL,
source=None)
euler_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
from hedge.timestep import RK4TimeStepper
stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_name = ("euler-sinewave-%(order)d-%(els)d.dat" % {
"order": order,
"els": len(mesh.elements)
})
else:
log_name = False
logmgr = LogManager(log_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time,
logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(
discr, stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
for step, t, dt in step_it:
#if step % 10 == 0:
if write_output:
visf = vis.make_file("sinewave-%d-%04d" % (order, step))
#from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(
visf,
[
("rho",
discr.convert_volume(op.rho(fields),
kind="numpy")),
("e",
discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u",
discr.convert_volume(op.rho_u(fields),
kind="numpy")),
("u",
discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", op.rho(true_fields)),
#("true_e", op.e(true_fields)),
#("true_rho_u", op.rho_u(true_fields)),
#("true_u", op.u(true_fields)),
#("rhs_rho", op.rho(rhs_fields)),
#("rhs_e", op.e(rhs_fields)),
#("rhs_rho_u", op.rho_u(rhs_fields)),
],
#expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
#("p", "0.4*(e- 0.5*(rho_u*u))"),
#],
time=t,
step=step)
visf.close()
fields = stepper(fields, t, dt, rhs)
finally:
vis.close()
logmgr.close()
discr.close()
true_fields = sinewave.volume_interpolant(t, discr)
eoc_rec.add_data_point(order, discr.norm(fields - true_fields))
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
def test_elliptic():
"""Test various properties of elliptic operators."""
from hedge.tools import unit_vector
def matrix_rep(op):
h, w = op.shape
mat = numpy.zeros(op.shape)
for j in range(w):
mat[:, j] = op(unit_vector(w, j))
return mat
def check_grad_mat():
import pyublas
if not pyublas.has_sparse_wrappers():
return
grad_mat = op.grad_matrix()
#print len(discr), grad_mat.nnz, type(grad_mat)
for i in range(10):
u = numpy.random.randn(len(discr))
mat_result = grad_mat * u
op_result = numpy.hstack(op.grad(u))
err = la.norm(mat_result - op_result) * la.norm(op_result)
assert err < 1e-5
def check_matrix_tgt():
big = numpy.zeros((20, 20), flavor=numpy.SparseBuildMatrix)
small = numpy.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print small
from hedge._internal import MatrixTarget
tgt = MatrixTarget(big, 4, 4)
tgt.begin(small.shape[0], small.shape[1])
print "YO"
tgt.add_coefficients(4, 4, small)
print "DUDE"
tgt.finalize()
print big
import pymbolic
v_x = pymbolic.var("x")
truesol = pymbolic.parse("math.sin(x[0]**2*x[1]**2)")
truesol_c = pymbolic.compile(truesol, variables=["x"])
def laplace(expression, variables):
return sum(
pymbolic.diff(pymbolic.diff(expression, var), var)
for var in variables)
rhs = laplace(truesol, [v_x[0], v_x[1]])
rhs_c = pymbolic.compile(rhs, variables=["x", "el"])
from hedge.mesh import TAG_ALL, TAG_NONE
from hedge.mesh.generator import make_disk_mesh
mesh = make_disk_mesh(r=0.5, max_area=0.1, faces=20)
mesh = mesh.reordered_by("cuthill")
from hedge.backends import CPURunContext
rcon = CPURunContext()
from hedge.tools import EOCRecorder
eocrec = EOCRecorder()
for order in [1, 2, 3, 4, 5]:
for flux in ["ldg", "ip"]:
from hedge.discretization.local import TriangleDiscretization
discr = rcon.make_discretization(
mesh,
TriangleDiscretization(order),
debug=discr_class.noninteractive_debug_flags())
from hedge.data import GivenFunction
from hedge.models.poisson import PoissonOperator
op = PoissonOperator(
discr.dimensions,
dirichlet_tag=TAG_ALL,
dirichlet_bc=GivenFunction(lambda x, el: truesol_c(x)),
neumann_tag=TAG_NONE)
bound_op = op.bind(discr)
if order <= 3:
mat = matrix_rep(bound_op)
sym_err = la.norm(mat - mat.T)
#print sym_err
assert sym_err < 1e-12
#check_grad_mat()
from hedge.iterative import parallel_cg
truesol_v = discr.interpolate_volume_function(
lambda x, el: truesol_c(x))
sol_v = -parallel_cg(rcon,
-bound_op,
bound_op.prepare_rhs(
discr.interpolate_volume_function(rhs_c)),
tol=1e-10,
max_iterations=40000)
eocrec.add_data_point(order, discr.norm(sol_v - truesol_v))
#print eocrec.pretty_print()
assert eocrec.estimate_order_of_convergence()[0, 1] > 8
def main(write_output=True):
from hedge.backends import guess_run_context
rcon = guess_run_context(
#["cuda"]
)
gamma = 1.4
# at A=1 we have case of isentropic vortex, source terms
# arise for other values
densityA = 2.0
from hedge.tools import EOCRecorder, to_obj_array
eoc_rec = EOCRecorder()
if rcon.is_head_rank:
from hedge.mesh import \
make_rect_mesh, \
make_centered_regular_rect_mesh
refine = 1
mesh = make_centered_regular_rect_mesh((0,-5), (10,5), n=(9,9),
post_refine_factor=refine)
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [4,5]:
discr = rcon.make_discretization(mesh_data, order=order,
debug=[#"cuda_no_plan",
#"print_op_code"
],
default_scalar_type=numpy.float64)
from hedge.visualization import SiloVisualizer, VtkVisualizer
#vis = VtkVisualizer(discr, rcon, "vortex-%d" % order)
vis = SiloVisualizer(discr, rcon)
vortex = Vortex(beta=5, gamma=gamma,
center=[5,0],
velocity=[1,0], densityA=densityA)
fields = vortex.volume_interpolant(0, discr)
sources=SourceTerms(beta=5, gamma=gamma,
center=[5,0],
velocity=[1,0], densityA=densityA)
from hedge.models.gas_dynamics import (
GasDynamicsOperator, GammaLawEOS)
from hedge.mesh import TAG_ALL
op = GasDynamicsOperator(dimensions=2,
mu=0.0, prandtl=0.72, spec_gas_const=287.1,
equation_of_state=GammaLawEOS(vortex.gamma),
bc_inflow=vortex, bc_outflow=vortex, bc_noslip=vortex,
inflow_tag=TAG_ALL, source=sources)
euler_ex = op.bind(discr)
max_eigval = [0]
def rhs(t, q):
ode_rhs, speed = euler_ex(t, q)
max_eigval[0] = speed
return ode_rhs
rhs(0, fields)
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
# limiter setup -------------------------------------------------------
from hedge.models.gas_dynamics import SlopeLimiter1NEuler
limiter = SlopeLimiter1NEuler(discr, gamma, 2, op)
# time stepper --------------------------------------------------------
from hedge.timestep import SSPRK3TimeStepper, RK4TimeStepper
#stepper = SSPRK3TimeStepper(limiter=limiter)
#stepper = SSPRK3TimeStepper()
stepper = RK4TimeStepper()
# diagnostics setup ---------------------------------------------------
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_file_name = "euler-%d.dat" % order
else:
log_file_name = None
logmgr = LogManager(log_file_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"])
# timestep loop -------------------------------------------------------
t = 0
#fields = limiter(fields)
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=.1,
#max_steps=500,
logmgr=logmgr,
max_dt_getter=lambda t: 0.4*op.estimate_timestep(discr,
stepper=stepper, t=t, max_eigenvalue=max_eigval[0]))
for step, t, dt in step_it:
if step % 1 == 0 and write_output:
#if False:
visf = vis.make_file("vortex-%d-%04d" % (order, step))
true_fields = vortex.volume_interpolant(t, discr)
#rhs_fields = rhs(t, fields)
from pyvisfile.silo import DB_VARTYPE_VECTOR
vis.add_data(visf,
[
("rho", discr.convert_volume(op.rho(fields), kind="numpy")),
("e", discr.convert_volume(op.e(fields), kind="numpy")),
("rho_u", discr.convert_volume(op.rho_u(fields), kind="numpy")),
("u", discr.convert_volume(op.u(fields), kind="numpy")),
#("true_rho", discr.convert_volume(op.rho(true_fields), kind="numpy")),
#("true_e", discr.convert_volume(op.e(true_fields), kind="numpy")),
#("true_rho_u", discr.convert_volume(op.rho_u(true_fields), kind="numpy")),
#("true_u", discr.convert_volume(op.u(true_fields), kind="numpy")),
#("rhs_rho", discr.convert_volume(op.rho(rhs_fields), kind="numpy")),
#("rhs_e", discr.convert_volume(op.e(rhs_fields), kind="numpy")),
#("rhs_rho_u", discr.convert_volume(op.rho_u(rhs_fields), kind="numpy")),
],
expressions=[
#("diff_rho", "rho-true_rho"),
#("diff_e", "e-true_e"),
#("diff_rho_u", "rho_u-true_rho_u", DB_VARTYPE_VECTOR),
("p", "0.4*(e- 0.5*(rho_u*u))"),
],
time=t, step=step
)
visf.close()
fields = stepper(fields, t, dt, rhs)
true_fields = vortex.volume_interpolant(t, discr)
l2_error = discr.norm(fields-true_fields)
l2_error_rho = discr.norm(op.rho(fields)-op.rho(true_fields))
l2_error_e = discr.norm(op.e(fields)-op.e(true_fields))
l2_error_rhou = discr.norm(op.rho_u(fields)-op.rho_u(true_fields))
l2_error_u = discr.norm(op.u(fields)-op.u(true_fields))
eoc_rec.add_data_point(order, l2_error_rho)
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
logmgr.set_constant("l2_error", l2_error)
logmgr.set_constant("l2_error_rho", l2_error_rho)
logmgr.set_constant("l2_error_e", l2_error_e)
logmgr.set_constant("l2_error_rhou", l2_error_rhou)
logmgr.set_constant("l2_error_u", l2_error_u)
logmgr.set_constant("refinement", refine)
finally:
if write_output:
vis.close()
logmgr.close()
discr.close()
def main(write_output=True,
allow_features=None,
flux_type_arg=1,
bdry_flux_type_arg=None,
extra_discr_args={}):
from hedge.mesh.generator import make_cylinder_mesh, make_box_mesh
from hedge.tools import EOCRecorder, to_obj_array
from math import sqrt, pi # noqa
from analytic_solutions import ( # noqa
RealPartAdapter, SplitComplexAdapter, CylindricalFieldAdapter,
CylindricalCavityMode, RectangularWaveguideMode, RectangularCavityMode)
from hedge.models.em import MaxwellOperator
logging.basicConfig(level=logging.DEBUG)
from hedge.backends import guess_run_context
rcon = guess_run_context(allow_features)
epsilon0 = 8.8541878176e-12 # C**2 / (N m**2)
mu0 = 4 * pi * 1e-7 # N/A**2.
epsilon = 1 * epsilon0
mu = 1 * mu0
eoc_rec = EOCRecorder()
cylindrical = False
periodic = False
if cylindrical:
R = 1
d = 2
mode = CylindricalCavityMode(m=1,
n=1,
p=1,
radius=R,
height=d,
epsilon=epsilon,
mu=mu)
# r_sol = CylindricalFieldAdapter(RealPartAdapter(mode))
# c_sol = SplitComplexAdapter(CylindricalFieldAdapter(mode))
if rcon.is_head_rank:
mesh = make_cylinder_mesh(radius=R, height=d, max_volume=0.01)
else:
if periodic:
mode = RectangularWaveguideMode(epsilon, mu, (3, 2, 1))
periodicity = (False, False, True)
else:
periodicity = None
mode = RectangularCavityMode(epsilon, mu, (1, 2, 2))
if rcon.is_head_rank:
mesh = make_box_mesh(max_volume=0.001, periodicity=periodicity)
if rcon.is_head_rank:
mesh_data = rcon.distribute_mesh(mesh)
else:
mesh_data = rcon.receive_mesh()
for order in [4, 5, 6]:
#for order in [1,2,3,4,5,6]:
extra_discr_args.setdefault("debug", []).extend(
["cuda_no_plan", "cuda_dump_kernels"])
op = MaxwellOperator(epsilon,
mu,
flux_type=flux_type_arg,
bdry_flux_type=bdry_flux_type_arg)
discr = rcon.make_discretization(mesh_data,
order=order,
tune_for=op.op_template(),
**extra_discr_args)
from hedge.visualization import VtkVisualizer
if write_output:
vis = VtkVisualizer(discr, rcon, "em-%d" % order)
mode.set_time(0)
def get_true_field():
return discr.convert_volume(to_obj_array(
mode(discr).real.astype(discr.default_scalar_type).copy()),
kind=discr.compute_kind)
fields = get_true_field()
if rcon.is_head_rank:
print "---------------------------------------------"
print "order %d" % order
print "---------------------------------------------"
print "#elements=", len(mesh.elements)
from hedge.timestep.runge_kutta import LSRK4TimeStepper
stepper = LSRK4TimeStepper(dtype=discr.default_scalar_type, rcon=rcon)
#from hedge.timestep.dumka3 import Dumka3TimeStepper
#stepper = Dumka3TimeStepper(3, dtype=discr.default_scalar_type, rcon=rcon)
# {{{ diagnostics setup
from pytools.log import LogManager, add_general_quantities, \
add_simulation_quantities, add_run_info
if write_output:
log_file_name = "maxwell-%d.dat" % order
else:
log_file_name = None
logmgr = LogManager(log_file_name, "w", rcon.communicator)
add_run_info(logmgr)
add_general_quantities(logmgr)
add_simulation_quantities(logmgr)
discr.add_instrumentation(logmgr)
stepper.add_instrumentation(logmgr)
from pytools.log import IntervalTimer
vis_timer = IntervalTimer("t_vis", "Time spent visualizing")
logmgr.add_quantity(vis_timer)
from hedge.log import EMFieldGetter, add_em_quantities
field_getter = EMFieldGetter(discr, op, lambda: fields)
add_em_quantities(logmgr, op, field_getter)
logmgr.add_watches(
["step.max", "t_sim.max", ("W_field", "W_el+W_mag"), "t_step.max"])
# }}}
# {{{ timestep loop
rhs = op.bind(discr)
final_time = 0.5e-9
try:
from hedge.timestep import times_and_steps
step_it = times_and_steps(
final_time=final_time,
logmgr=logmgr,
max_dt_getter=lambda t: op.estimate_timestep(
discr, stepper=stepper, t=t, fields=fields))
for step, t, dt in step_it:
if step % 50 == 0 and write_output:
sub_timer = vis_timer.start_sub_timer()
e, h = op.split_eh(fields)
visf = vis.make_file("em-%d-%04d" % (order, step))
vis.add_data(visf, [
("e", discr.convert_volume(e, kind="numpy")),
("h", discr.convert_volume(h, kind="numpy")),
],
time=t,
step=step)
visf.close()
sub_timer.stop().submit()
fields = stepper(fields, t, dt, rhs)
mode.set_time(final_time)
eoc_rec.add_data_point(order,
discr.norm(fields - get_true_field()))
finally:
if write_output:
vis.close()
logmgr.close()
discr.close()
if rcon.is_head_rank:
print
print eoc_rec.pretty_print("P.Deg.", "L2 Error")
# }}}
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 6