def test_kv_with_no_charge():
from random import seed
seed(0)
from pyrticle.units import SIUnitsWithNaturalConstants
units = SIUnitsWithNaturalConstants()
# discretization setup ----------------------------------------------------
from hedge.mesh import make_cylinder_mesh
from hedge.backends import guess_run_context
rcon = guess_run_context([])
tube_length = 100 * units.MM
mesh = make_cylinder_mesh(radius=25 * units.MM,
height=tube_length,
periodic=True)
discr = rcon.make_discretization(mesh, order=3)
dt = discr.dt_factor(units.VACUUM_LIGHT_SPEED()) / 2
final_time = 1 * units.M / units.VACUUM_LIGHT_SPEED()
nsteps = int(final_time / dt) + 1
dt = final_time / nsteps
# particles setup ---------------------------------------------------------
from pyrticle.cloud import PicMethod
from pyrticle.deposition.shape import ShapeFunctionDepositor
from pyrticle.pusher import MonomialParticlePusher
method = PicMethod(discr, units, ShapeFunctionDepositor(),
MonomialParticlePusher(), 3, 3)
nparticles = 10000
cloud_charge = 1e-9 * units.C
electrons_per_particle = cloud_charge / nparticles / units.EL_CHARGE
el_energy = 5.2e6 * units.EV
gamma = el_energy / units.EL_REST_ENERGY()
beta = (1 - 1 / gamma**2)**0.5
from pyrticle.distribution import KVZIntervalBeam
beam = KVZIntervalBeam(units,
total_charge=0,
p_charge=0,
p_mass=electrons_per_particle * units.EL_MASS,
radii=2 * [2.5 * units.MM],
emittances=2 * [5 * units.MM * units.MRAD],
z_length=5 * units.MM,
z_pos=10 * units.MM,
beta=beta)
state = method.make_state()
method.add_particles(state, beam.generate_particles(), nparticles)
# diagnostics setup -------------------------------------------------------
from pytools.log import LogManager
from pyrticle.log import add_beam_quantities, StateObserver
observer = StateObserver(method, None)
logmgr = LogManager(mode="w")
add_beam_quantities(logmgr, observer, axis=0, beam_axis=2)
from pyrticle.distribution import KVPredictedRadius
logmgr.add_quantity(
KVPredictedRadius(dt,
beam_v=beta * units.VACUUM_LIGHT_SPEED(),
predictor=beam.get_rms_predictor(axis=0),
suffix="x_rms"))
logmgr.add_quantity(
KVPredictedRadius(dt,
beam_v=beta * units.VACUUM_LIGHT_SPEED(),
predictor=beam.get_total_predictor(axis=0),
suffix="x_total"))
# timestep loop -----------------------------------------------------------
vel = method.velocities(state)
from hedge.tools import join_fields
def rhs(t, y):
return join_fields([
vel,
0 * vel,
0, # drecon
])
from hedge.timestep.runge_kutta import LSRK4TimeStepper
stepper = LSRK4TimeStepper()
t = 0
from pyrticle.cloud import TimesteppablePicState
ts_state = TimesteppablePicState(method, state)
for step in xrange(nsteps):
observer.set_fields_and_state(None, ts_state.state)
logmgr.tick()
ts_state = stepper(ts_state, t, dt, rhs)
method.upkeep(ts_state.state)
t += dt
logmgr.tick()
_, _, err_table = logmgr.get_expr_dataset(
"(rx_rms-rx_rms_theory)/rx_rms_theory")
rel_max_rms_error = max(err for step, err in err_table)
assert rel_max_rms_error < 0.01
def test_efield_vs_gauss_law():
from hedge.mesh.generator import \
make_box_mesh, \
make_cylinder_mesh
from math import sqrt, pi
from pytools.arithmetic_container import \
ArithmeticList, join_fields
from random import seed
from pytools.stopwatch import Job
from pyrticle.units import SIUnitsWithNaturalConstants
units = SIUnitsWithNaturalConstants()
seed(0)
nparticles = 10000
beam_radius = 2.5 * units.MM
emittance = 5 * units.MM * units.MRAD
final_time = 0.1 * units.M / units.VACUUM_LIGHT_SPEED()
field_dump_interval = 1
tube_length = 20 * units.MM
# discretization setup ----------------------------------------------------
from pyrticle.geometry import make_cylinder_with_fine_core
mesh = make_cylinder_with_fine_core(
r=10 * beam_radius,
inner_r=1 * beam_radius,
min_z=0,
max_z=tube_length,
max_volume_inner=10 * units.MM**3,
max_volume_outer=100 * units.MM**3,
radial_subdiv=10,
)
from hedge.backends import guess_run_context
rcon = guess_run_context([])
discr = rcon.make_discretization(mesh, order=3)
from hedge.models.em import MaxwellOperator
max_op = MaxwellOperator(epsilon=units.EPSILON0, mu=units.MU0, flux_type=1)
from hedge.models.nd_calculus import DivergenceOperator
div_op = DivergenceOperator(discr.dimensions)
# particles setup ---------------------------------------------------------
from pyrticle.cloud import PicMethod
from pyrticle.deposition.shape import ShapeFunctionDepositor
from pyrticle.pusher import MonomialParticlePusher
method = PicMethod(discr, units, ShapeFunctionDepositor(),
MonomialParticlePusher(), 3, 3)
# particle ic ---------------------------------------------------------
cloud_charge = -1e-9 * units.C
electrons_per_particle = abs(cloud_charge / nparticles / units.EL_CHARGE)
el_energy = 10 * units.EL_REST_ENERGY()
el_lorentz_gamma = el_energy / units.EL_REST_ENERGY()
beta = (1 - 1 / el_lorentz_gamma**2)**0.5
gamma = 1 / sqrt(1 - beta**2)
from pyrticle.distribution import KVZIntervalBeam
beam = KVZIntervalBeam(units,
total_charge=cloud_charge,
p_charge=cloud_charge / nparticles,
p_mass=electrons_per_particle * units.EL_MASS,
radii=2 * [beam_radius],
emittances=2 * [5 * units.MM * units.MRAD],
z_length=tube_length,
z_pos=tube_length / 2,
beta=beta)
state = method.make_state()
method.add_particles(state, beam.generate_particles(), nparticles)
# field ic ----------------------------------------------------------------
from pyrticle.cloud import guess_shape_bandwidth
guess_shape_bandwidth(method, state, 2)
from pyrticle.cloud import compute_initial_condition
from hedge.data import ConstantGivenFunction
fields = compute_initial_condition(rcon,
discr,
method,
state,
maxwell_op=max_op,
potential_bc=ConstantGivenFunction())
# check against theory ----------------------------------------------------
q_per_unit_z = cloud_charge / beam.z_length
class TheoreticalEField:
shape = (3, )
def __call__(self, x, el):
r = la.norm(x[:2])
if r >= max(beam.radii):
xy_unit = x / r
xy_unit[2] = 0
return xy_unit * ((q_per_unit_z) /
(2 * pi * r * max_op.epsilon))
else:
return numpy.zeros((3, ))
def theory_indicator(x, el):
r = la.norm(x[:2])
if r >= max(beam.radii):
return 1
else:
return 0
from hedge.tools import join_fields, to_obj_array
e_theory = to_obj_array(
discr.interpolate_volume_function(TheoreticalEField()))
theory_ind = discr.interpolate_volume_function(theory_indicator)
e_field, h_field = max_op.split_eh(fields)
restricted_e = join_fields(*[e_i * theory_ind for e_i in e_field])
def l2_error(field, true):
return discr.norm(field - true) / discr.norm(true)
outer_l2 = l2_error(restricted_e, e_theory)
assert outer_l2 < 0.08
if False:
visf = vis.make_file("e_comparison")
mesh_scalars, mesh_vectors = \
method.add_to_vis(vis, visf)
vis.add_data(visf, [
("e", restricted_e),
("e_theory", e_theory),
] + mesh_vectors + mesh_scalars)
visf.close()
def test_kv_with_no_charge():
from random import seed
seed(0)
from pyrticle.units import SIUnitsWithNaturalConstants
units = SIUnitsWithNaturalConstants()
# discretization setup ----------------------------------------------------
from hedge.mesh import make_cylinder_mesh
from hedge.backends import guess_run_context
rcon = guess_run_context([])
tube_length = 100*units.MM
mesh = make_cylinder_mesh(radius=25*units.MM, height=tube_length, periodic=True)
discr = rcon.make_discretization(mesh, order=3)
dt = discr.dt_factor(units.VACUUM_LIGHT_SPEED()) / 2
final_time = 1*units.M/units.VACUUM_LIGHT_SPEED()
nsteps = int(final_time/dt)+1
dt = final_time/nsteps
# particles setup ---------------------------------------------------------
from pyrticle.cloud import PicMethod
from pyrticle.deposition.shape import ShapeFunctionDepositor
from pyrticle.pusher import MonomialParticlePusher
method = PicMethod(discr, units,
ShapeFunctionDepositor(),
MonomialParticlePusher(),
3, 3)
nparticles = 10000
cloud_charge = 1e-9 * units.C
electrons_per_particle = cloud_charge/nparticles/units.EL_CHARGE
el_energy = 5.2e6 * units.EV
gamma = el_energy/units.EL_REST_ENERGY()
beta = (1-1/gamma**2)**0.5
from pyrticle.distribution import KVZIntervalBeam
beam = KVZIntervalBeam(units,
total_charge=0,
p_charge=0,
p_mass=electrons_per_particle*units.EL_MASS,
radii=2*[2.5*units.MM],
emittances=2*[5 * units.MM * units.MRAD],
z_length=5*units.MM,
z_pos=10*units.MM,
beta=beta)
state = method.make_state()
method.add_particles(state, beam.generate_particles(), nparticles)
# diagnostics setup -------------------------------------------------------
from pytools.log import LogManager
from pyrticle.log import add_beam_quantities, StateObserver
observer = StateObserver(method, None)
logmgr = LogManager(mode="w")
add_beam_quantities(logmgr, observer, axis=0, beam_axis=2)
from pyrticle.distribution import KVPredictedRadius
logmgr.add_quantity(KVPredictedRadius(dt,
beam_v=beta*units.VACUUM_LIGHT_SPEED(),
predictor=beam.get_rms_predictor(axis=0),
suffix="x_rms"))
logmgr.add_quantity(KVPredictedRadius(dt,
beam_v=beta*units.VACUUM_LIGHT_SPEED(),
predictor=beam.get_total_predictor(axis=0),
suffix="x_total"))
# timestep loop -----------------------------------------------------------
vel = method.velocities(state)
from hedge.tools import join_fields
def rhs(t, y):
return join_fields([
vel,
0*vel,
0, # drecon
])
from hedge.timestep.runge_kutta import LSRK4TimeStepper
stepper = LSRK4TimeStepper()
t = 0
from pyrticle.cloud import TimesteppablePicState
ts_state = TimesteppablePicState(method, state)
for step in xrange(nsteps):
observer.set_fields_and_state(None, ts_state.state)
logmgr.tick()
ts_state = stepper(ts_state, t, dt, rhs)
method.upkeep(ts_state.state)
t += dt
logmgr.tick()
_, _, err_table = logmgr.get_expr_dataset("(rx_rms-rx_rms_theory)/rx_rms_theory")
rel_max_rms_error = max(err for step, err in err_table)
assert rel_max_rms_error < 0.01
def test_efield_vs_gauss_law():
from hedge.mesh.generator import \
make_box_mesh, \
make_cylinder_mesh
from math import sqrt, pi
from pytools.arithmetic_container import \
ArithmeticList, join_fields
from random import seed
from pytools.stopwatch import Job
from pyrticle.units import SIUnitsWithNaturalConstants
units = SIUnitsWithNaturalConstants()
seed(0)
nparticles = 10000
beam_radius = 2.5 * units.MM
emittance = 5 * units.MM * units.MRAD
final_time = 0.1*units.M/units.VACUUM_LIGHT_SPEED()
field_dump_interval = 1
tube_length = 20*units.MM
# discretization setup ----------------------------------------------------
from pyrticle.geometry import make_cylinder_with_fine_core
mesh = make_cylinder_with_fine_core(
r=10*beam_radius, inner_r=1*beam_radius,
min_z=0, max_z=tube_length,
max_volume_inner=10*units.MM**3,
max_volume_outer=100*units.MM**3,
radial_subdiv=10,
)
from hedge.backends import guess_run_context
rcon = guess_run_context([])
discr = rcon.make_discretization(mesh, order=3)
from hedge.models.em import MaxwellOperator
max_op = MaxwellOperator(
epsilon=units.EPSILON0,
mu=units.MU0,
flux_type=1)
from hedge.models.nd_calculus import DivergenceOperator
div_op = DivergenceOperator(discr.dimensions)
# particles setup ---------------------------------------------------------
from pyrticle.cloud import PicMethod
from pyrticle.deposition.shape import ShapeFunctionDepositor
from pyrticle.pusher import MonomialParticlePusher
method = PicMethod(discr, units,
ShapeFunctionDepositor(),
MonomialParticlePusher(),
3, 3)
# particle ic ---------------------------------------------------------
cloud_charge = -1e-9 * units.C
electrons_per_particle = abs(cloud_charge/nparticles/units.EL_CHARGE)
el_energy = 10*units.EL_REST_ENERGY()
el_lorentz_gamma = el_energy/units.EL_REST_ENERGY()
beta = (1-1/el_lorentz_gamma**2)**0.5
gamma = 1/sqrt(1-beta**2)
from pyrticle.distribution import KVZIntervalBeam
beam = KVZIntervalBeam(units, total_charge=cloud_charge,
p_charge=cloud_charge/nparticles,
p_mass=electrons_per_particle*units.EL_MASS,
radii=2*[beam_radius],
emittances=2*[5 * units.MM * units.MRAD],
z_length=tube_length,
z_pos=tube_length/2,
beta=beta)
state = method.make_state()
method.add_particles(state, beam.generate_particles(), nparticles)
# field ic ----------------------------------------------------------------
from pyrticle.cloud import guess_shape_bandwidth
guess_shape_bandwidth(method, state, 2)
from pyrticle.cloud import compute_initial_condition
from hedge.data import ConstantGivenFunction
fields = compute_initial_condition(
rcon,
discr, method, state, maxwell_op=max_op,
potential_bc=ConstantGivenFunction())
# check against theory ----------------------------------------------------
q_per_unit_z = cloud_charge/beam.z_length
class TheoreticalEField:
shape = (3,)
def __call__(self, x, el):
r = la.norm(x[:2])
if r >= max(beam.radii):
xy_unit = x/r
xy_unit[2] = 0
return xy_unit*((q_per_unit_z)
/
(2*pi*r*max_op.epsilon))
else:
return numpy.zeros((3,))
def theory_indicator(x, el):
r = la.norm(x[:2])
if r >= max(beam.radii):
return 1
else:
return 0
from hedge.tools import join_fields, to_obj_array
e_theory = to_obj_array(discr.interpolate_volume_function(TheoreticalEField()))
theory_ind = discr.interpolate_volume_function(theory_indicator)
e_field, h_field = max_op.split_eh(fields)
restricted_e = join_fields(*[e_i * theory_ind for e_i in e_field])
def l2_error(field, true):
return discr.norm(field-true)/discr.norm(true)
outer_l2 = l2_error(restricted_e, e_theory)
assert outer_l2 < 0.08
if False:
visf = vis.make_file("e_comparison")
mesh_scalars, mesh_vectors = \
method.add_to_vis(vis, visf)
vis.add_data(visf, [
("e", restricted_e),
("e_theory", e_theory),
]
+ mesh_vectors
+ mesh_scalars
)
visf.close()
def run_setup(units, casename, setup, discr, pusher, visualize=False):
from hedge.timestep.runge_kutta import LSRK4TimeStepper
from hedge.visualization import SiloVisualizer
from hedge.models.em import MaxwellOperator
vis = SiloVisualizer(discr)
from pyrticle.cloud import PicMethod
from pyrticle.deposition.shape import ShapeFunctionDepositor
method = PicMethod(discr,
units,
ShapeFunctionDepositor(),
pusher(),
3,
3,
debug=set(["verbose_vis"]))
e, h = setup.fields(discr)
b = units.MU0 * h
init_positions = setup.positions(0)
init_velocities = setup.velocities(0)
nparticles = len(init_positions)
state = method.make_state()
method.add_particles(
state,
zip(
init_positions,
init_velocities,
nparticles * [setup.charge],
nparticles * [units.EL_MASS],
), nparticles)
final_time = setup.final_time()
nsteps = setup.nsteps()
dt = final_time / nsteps
# timestepping ------------------------------------------------------------
from hedge.models.em import MaxwellOperator
max_op = MaxwellOperator(epsilon=units.EPSILON0, mu=units.MU0, flux_type=1)
fields = max_op.assemble_eh(e, h)
from pyrticle.cloud import \
FieldToParticleRhsCalculator, \
ParticleRhsCalculator
p_rhs_calculator = ParticleRhsCalculator(method, max_op)
f2p_rhs_calculator = FieldToParticleRhsCalculator(method, max_op)
def rhs(t, ts_state):
return (p_rhs_calculator(t, lambda: fields, lambda: ts_state.state) +
f2p_rhs_calculator(t, lambda: fields, lambda: ts_state.state))
stepper = LSRK4TimeStepper()
t = 0
bbox = discr.mesh.bounding_box()
z_period = bbox[1][2] - bbox[0][2]
def check_result():
from hedge.tools import cross
deriv_dt = 1e-12
dim = discr.dimensions
true_x = setup.positions(t)
true_v = setup.velocities(t)
true_f = [(p2 - p1) / (2 * deriv_dt) for p1, p2 in zip(
setup.momenta(t - deriv_dt), setup.momenta(t + deriv_dt))]
state = ts_state.state
from pyrticle.tools import NumberShiftableVector
vis_info = state.vis_listener.particle_vis_map
sim_x = state.positions
sim_v = method.velocities(state)
sim_f = NumberShiftableVector.unwrap(vis_info["mag_force"] +
vis_info["el_force"])
sim_el_f = NumberShiftableVector.unwrap(vis_info["el_force"])
sim_mag_f = NumberShiftableVector.unwrap(vis_info["mag_force"])
local_e = setup.e()
local_b = units.MU0 * setup.h()
x_err = 0
v_err = 0
f_err = 0
for i in range(len(state)):
#real_f = numpy.array(cross(sim_v, setup.charge*local_b)) + setup.charge*local_e
my_true_x = true_x[i]
my_true_x[2] = my_true_x[2] % z_period
if False and i == 0:
#print "particle %d" % i
print "pos:", la.norm(true_x[i] - sim_x[i]) / la.norm(
true_x[i])
#print "vel:", la.norm(true_v[i]-sim_v[i])/la.norm(true_v[i])
#print "force:", la.norm(true_f[i]-sim_f[i])/la.norm(true_f[i])
print "pos:", true_x[i], sim_x[i]
#print "vel:", true_v[i], sim_v[i]
#print "force:", true_f[i], sim_f[i]
#print "forces%d:..." % i, sim_el_f[i], sim_mag_f[i]
#print "acc%d:" % i, la.norm(a-sim_a)
#u = numpy.vstack((v, sim_v, f, sim_f, real_f))
#print "acc%d:\n%s" % (i, u)
#raw_input()
def rel_err(sim, true):
return la.norm(true - sim) / la.norm(true)
x_err = max(x_err, rel_err(sim_x[i], my_true_x))
v_err = max(v_err, rel_err(sim_v[i], true_v[i]))
f_err = max(f_err, rel_err(sim_f[i], true_f[i]))
return x_err, v_err, f_err
# make sure verbose-vis fields are filled
from pyrticle.cloud import TimesteppablePicState
ts_state = TimesteppablePicState(method, state)
del state
rhs(t, ts_state)
errors = (0, 0, 0)
for step in xrange(nsteps):
if step % int(setup.nsteps() / 300) == 0:
errors = tuple(
max(old_err, new_err)
for old_err, new_err in zip(errors, check_result()))
if visualize:
visf = vis.make_file("%s-%04d" % (casename, step))
method.add_to_vis(vis, visf, ts_state.state, time=t, step=step)
if True:
vis.add_data(visf, [
("e", e),
("h", h),
],
time=t,
step=step)
else:
vis.add_data(visf, [], time=t, step=step)
visf.close()
method.upkeep(ts_state.state)
ts_state = stepper(ts_state, t, dt, rhs)
t += dt
assert errors[0] < 2e-12, casename + "-pos"
assert errors[1] < 2e-13, casename + "-vel"
assert errors[2] < 2e-4, casename + "-acc"
vis.close()
def run_setup(units, casename, setup, discr, pusher, visualize=False):
from hedge.timestep.runge_kutta import LSRK4TimeStepper
from hedge.visualization import SiloVisualizer
from hedge.models.em import MaxwellOperator
vis = SiloVisualizer(discr)
from pyrticle.cloud import PicMethod
from pyrticle.deposition.shape import ShapeFunctionDepositor
method = PicMethod(discr, units,
ShapeFunctionDepositor(),
pusher(),
3, 3, debug=set(["verbose_vis"]))
e, h = setup.fields(discr)
b = units.MU0 * h
init_positions = setup.positions(0)
init_velocities = setup.velocities(0)
nparticles = len(init_positions)
state = method.make_state()
method.add_particles(state,
zip(init_positions, init_velocities,
nparticles * [setup.charge],
nparticles * [units.EL_MASS],
),
nparticles)
final_time = setup.final_time()
nsteps = setup.nsteps()
dt = final_time/nsteps
# timestepping ------------------------------------------------------------
from hedge.models.em import MaxwellOperator
max_op = MaxwellOperator(
epsilon=units.EPSILON0,
mu=units.MU0,
flux_type=1)
fields = max_op.assemble_eh(e, h)
from pyrticle.cloud import \
FieldToParticleRhsCalculator, \
ParticleRhsCalculator
p_rhs_calculator = ParticleRhsCalculator(method, max_op)
f2p_rhs_calculator = FieldToParticleRhsCalculator(method, max_op)
def rhs(t, ts_state):
return (p_rhs_calculator(t, lambda: fields, lambda: ts_state.state)
+ f2p_rhs_calculator(t, lambda: fields, lambda: ts_state.state))
stepper = LSRK4TimeStepper()
t = 0
bbox = discr.mesh.bounding_box()
z_period = bbox[1][2] - bbox[0][2]
def check_result():
from hedge.tools import cross
deriv_dt = 1e-12
dim = discr.dimensions
true_x = setup.positions(t)
true_v = setup.velocities(t)
true_f = [(p2-p1)/(2*deriv_dt)
for p1, p2 in zip(setup.momenta(t-deriv_dt), setup.momenta(t+deriv_dt))]
state = ts_state.state
from pyrticle.tools import NumberShiftableVector
vis_info = state.vis_listener.particle_vis_map
sim_x = state.positions
sim_v = method.velocities(state)
sim_f = NumberShiftableVector.unwrap(
vis_info["mag_force"] + vis_info["el_force"])
sim_el_f = NumberShiftableVector.unwrap(vis_info["el_force"])
sim_mag_f = NumberShiftableVector.unwrap(vis_info["mag_force"])
local_e = setup.e()
local_b = units.MU0 * setup.h()
x_err = 0
v_err = 0
f_err = 0
for i in range(len(state)):
#real_f = numpy.array(cross(sim_v, setup.charge*local_b)) + setup.charge*local_e
my_true_x = true_x[i]
my_true_x[2] = my_true_x[2] % z_period
if False and i == 0:
#print "particle %d" % i
print "pos:", la.norm(true_x[i]-sim_x[i])/la.norm(true_x[i])
#print "vel:", la.norm(true_v[i]-sim_v[i])/la.norm(true_v[i])
#print "force:", la.norm(true_f[i]-sim_f[i])/la.norm(true_f[i])
print "pos:", true_x[i], sim_x[i]
#print "vel:", true_v[i], sim_v[i]
#print "force:", true_f[i], sim_f[i]
#print "forces%d:..." % i, sim_el_f[i], sim_mag_f[i]
#print "acc%d:" % i, la.norm(a-sim_a)
#u = numpy.vstack((v, sim_v, f, sim_f, real_f))
#print "acc%d:\n%s" % (i, u)
#raw_input()
def rel_err(sim, true):
return la.norm(true-sim)/la.norm(true)
x_err = max(x_err, rel_err(sim_x[i], my_true_x))
v_err = max(v_err, rel_err(sim_v[i], true_v[i]))
f_err = max(f_err, rel_err(sim_f[i], true_f[i]))
return x_err, v_err, f_err
# make sure verbose-vis fields are filled
from pyrticle.cloud import TimesteppablePicState
ts_state = TimesteppablePicState(method, state)
del state
rhs(t, ts_state)
errors = (0, 0, 0)
for step in xrange(nsteps):
if step % int(setup.nsteps()/300) == 0:
errors = tuple(
max(old_err, new_err)
for old_err, new_err in zip(errors, check_result()))
if visualize:
visf = vis.make_file("%s-%04d" % (casename, step))
method.add_to_vis(vis, visf, ts_state.state, time=t, step=step)
if True:
vis.add_data(visf, [ ("e", e), ("h", h), ],
time=t, step=step)
else:
vis.add_data(visf, [], time=t, step=step)
visf.close()
method.upkeep(ts_state.state)
ts_state = stepper(ts_state, t, dt, rhs)
t += dt
assert errors[0] < 2e-12, casename+"-pos"
assert errors[1] < 2e-13, casename+"-vel"
assert errors[2] < 2e-4, casename+"-acc"
vis.close()