def grad_operator(discr, dd_vol, dd_faces, u, flux):
r"""Compute a DG gradient for the input *u* with flux given by *flux*.
Parameters
----------
discr: grudge.eager.EagerDGDiscretization
the discretization to use
dd_vol: grudge.dof_desc.DOFDesc
the degree-of-freedom tag associated with the volume discrezation.
This determines the type of quadrature to be used.
dd_faces: grudge.dof_desc.DOFDesc
the degree-of-freedom tag associated with the surface discrezation.
This determines the type of quadrature to be used.
u: meshmode.dof_array.DOFArray or numpy.ndarray
the function (or container of functions) for which gradient is to be
calculated
flux: numpy.ndarray
the boundary flux across the faces of the element for each component
of *u*
Returns
-------
meshmode.dof_array.DOFArray or numpy.ndarray
the dg gradient operator applied to *u*
"""
return -discr.inverse_mass(
op.weak_local_grad(discr, dd_vol, u) -
op.face_mass(discr, dd_faces, flux))
def wave_operator(dcoll, c, w):
u = w[0]
v = w[1:]
dir_u = op.project(dcoll, "vol", BTAG_ALL, u)
dir_v = op.project(dcoll, "vol", BTAG_ALL, v)
dir_bval = flat_obj_array(dir_u, dir_v)
dir_bc = flat_obj_array(-dir_u, dir_v)
dd_quad = DOFDesc("vol", DISCR_TAG_QUAD)
c_quad = op.project(dcoll, "vol", dd_quad, c)
w_quad = op.project(dcoll, "vol", dd_quad, w)
u_quad = w_quad[0]
v_quad = w_quad[1:]
dd_allfaces_quad = DOFDesc("all_faces", DISCR_TAG_QUAD)
return (op.inverse_mass(
dcoll,
flat_obj_array(-op.weak_local_div(dcoll, dd_quad, c_quad * v_quad),
-op.weak_local_grad(dcoll, dd_quad, c_quad * u_quad))
+ # noqa: W504
op.face_mass(
dcoll, dd_allfaces_quad,
wave_flux(dcoll, c=c, w_tpair=op.interior_trace_pair(dcoll, w)) +
wave_flux(dcoll,
c=c,
w_tpair=TracePair(
BTAG_ALL, interior=dir_bval, exterior=dir_bc)))))
def operator(self, t, w):
dcoll = self.dcoll
u = w[0]
v = w[1:]
actx = u.array_context
# boundary conditions -------------------------------------------------
# dirichlet BCs -------------------------------------------------------
dir_u = op.project(dcoll, "vol", self.dirichlet_tag, u)
dir_v = op.project(dcoll, "vol", self.dirichlet_tag, v)
if self.dirichlet_bc_f:
# FIXME
from warnings import warn
warn("Inhomogeneous Dirichlet conditions on the wave equation "
"are still having issues.")
dir_g = self.dirichlet_bc_f
dir_bc = flat_obj_array(2 * dir_g - dir_u, dir_v)
else:
dir_bc = flat_obj_array(-dir_u, dir_v)
# neumann BCs ---------------------------------------------------------
neu_u = op.project(dcoll, "vol", self.neumann_tag, u)
neu_v = op.project(dcoll, "vol", self.neumann_tag, v)
neu_bc = flat_obj_array(neu_u, -neu_v)
# radiation BCs -------------------------------------------------------
rad_normal = thaw(dcoll.normal(dd=self.radiation_tag), actx)
rad_u = op.project(dcoll, "vol", self.radiation_tag, u)
rad_v = op.project(dcoll, "vol", self.radiation_tag, v)
rad_bc = flat_obj_array(
0.5 * (rad_u - self.sign * np.dot(rad_normal, rad_v)),
0.5 * rad_normal * (np.dot(rad_normal, rad_v) - self.sign * rad_u))
# entire operator -----------------------------------------------------
def flux(tpair):
return op.project(dcoll, tpair.dd, "all_faces", self.flux(tpair))
result = (op.inverse_mass(
dcoll,
flat_obj_array(-self.c * op.weak_local_div(dcoll, v),
-self.c * op.weak_local_grad(dcoll, u)) -
op.face_mass(
dcoll,
sum(
flux(tpair)
for tpair in op.interior_trace_pairs(dcoll, w)) +
flux(op.bv_trace_pair(dcoll, self.dirichlet_tag, w, dir_bc)) +
flux(op.bv_trace_pair(dcoll, self.neumann_tag, w, neu_bc)) +
flux(op.bv_trace_pair(dcoll, self.radiation_tag, w, rad_bc)))))
result[0] = result[0] + self.source_f(actx, dcoll, t)
return result
def grad_operator(discr, u, flux):
r"""Compute a DG gradient for the input *u* with flux given by *flux*.
Parameters
----------
discr: grudge.eager.EagerDGDiscretization
the discretization to use
u: meshmode.dof_array.DOFArray or numpy.ndarray
the DOF array (or object array of DOF arrays) to which the operator should be
applied
flux: numpy.ndarray
the boundary fluxes across the faces of the element
Returns
-------
meshmode.dof_array.DOFArray or numpy.ndarray
the dg gradient operator applied to *u*
"""
from grudge.op import weak_local_grad
return -discr.inverse_mass(
weak_local_grad(discr, u, nested=False) - discr.face_mass(flux))
def wave_operator(dcoll, c, w):
u = w[0]
v = w[1:]
dir_u = op.project(dcoll, "vol", BTAG_ALL, u)
dir_v = op.project(dcoll, "vol", BTAG_ALL, v)
dir_bval = flat_obj_array(dir_u, dir_v)
dir_bc = flat_obj_array(-dir_u, dir_v)
return (op.inverse_mass(
dcoll,
flat_obj_array(-c * op.weak_local_div(dcoll, v),
-c * op.weak_local_grad(dcoll, u)) + # noqa: W504
op.face_mass(
dcoll,
wave_flux(dcoll, c=c, w_tpair=op.interior_trace_pair(dcoll, w)) +
wave_flux(dcoll,
c=c,
w_tpair=TracePair(
BTAG_ALL, interior=dir_bval, exterior=dir_bc)))))
def wave_operator(dcoll, c, w):
u = w[0]
v = w[1:]
dir_u = op.project(dcoll, "vol", BTAG_ALL, u)
dir_v = op.project(dcoll, "vol", BTAG_ALL, v)
dir_bval = flat_obj_array(dir_u, dir_v)
dir_bc = flat_obj_array(-dir_u, dir_v)
dd_quad = DOFDesc("vol", DISCR_TAG_QUAD)
c_quad = op.project(dcoll, "vol", dd_quad, c)
w_quad = op.project(dcoll, "vol", dd_quad, w)
u_quad = w_quad[0]
v_quad = w_quad[1:]
dd_allfaces_quad = DOFDesc("all_faces", DISCR_TAG_QUAD)
return (
op.inverse_mass(
dcoll,
flat_obj_array(
-op.weak_local_div(dcoll, dd_quad, c_quad*v_quad),
-op.weak_local_grad(dcoll, dd_quad, c_quad*u_quad) \
# pylint: disable=invalid-unary-operand-type
) + op.face_mass(
dcoll,
dd_allfaces_quad,
wave_flux(
dcoll, c=c,
w_tpair=op.bdry_trace_pair(dcoll,
BTAG_ALL,
interior=dir_bval,
exterior=dir_bc)
) + sum(
wave_flux(dcoll, c=c, w_tpair=tpair)
for tpair in op.interior_trace_pairs(dcoll, w)
)
)
)
)
def wave_operator(dcoll, c, w):
u = w.u
v = w.v
dir_w = op.project(dcoll, "vol", BTAG_ALL, w)
dir_u = dir_w.u
dir_v = dir_w.v
dir_bval = WaveState(u=dir_u, v=dir_v)
dir_bc = WaveState(u=-dir_u, v=dir_v)
return (op.inverse_mass(
dcoll,
WaveState(u=-c * op.weak_local_div(dcoll, v),
v=-c * op.weak_local_grad(dcoll, u)) +
op.face_mass(
dcoll,
wave_flux(dcoll,
c=c,
w_tpair=op.bdry_trace_pair(
dcoll, BTAG_ALL, interior=dir_bval, exterior=dir_bc))
+ sum(
wave_flux(dcoll, c=c, w_tpair=tpair)
for tpair in op.interior_trace_pairs(dcoll, w)))))
def operator(self, t, u):
from meshmode.mesh import BTAG_ALL
dcoll = self.dcoll
def flux(tpair):
return op.project(dcoll, tpair.dd, "all_faces", self.flux(tpair))
if self.inflow_u is not None:
inflow_flux = flux(
op.bv_trace_pair(dcoll,
BTAG_ALL,
interior=u,
exterior=self.inflow_u(t)))
else:
inflow_flux = 0
return (op.inverse_mass(
dcoll,
np.dot(self.v, op.weak_local_grad(dcoll, u)) - op.face_mass(
dcoll,
sum(
flux(tpair)
for tpair in op.interior_trace_pairs(dcoll, u)) +
inflow_flux
# FIXME: Add support for inflow/outflow tags
# + flux(op.bv_trace_pair(dcoll,
# self.inflow_tag,
# interior=u,
# exterior=bc_in))
# + flux(op.bv_trace_pair(dcoll,
# self.outflow_tag,
# interior=u,
# exterior=bc_out))
)))
while t < t_final:
# extract the left boundary trace pair
lbnd_tpair = op.bv_trace_pair(dcoll,
dd=left_bndry,
interior=uh,
exterior=left_boundary_condition(x_bndry, t))
# extract the right boundary trace pair
rbnd_tpair = op.bv_trace_pair(dcoll,
dd=right_bndry,
interior=uh,
exterior=op.project(dcoll, "vol",
right_bndry, uh))
# extract the trace pairs on the interior faces
interior_tpair = op.interior_trace_pair(dcoll,
uh)
Su = op.weak_local_grad(dcoll, uh)
lift = op.face_mass(dcoll,
# left boundary weak-flux terms
op.project(dcoll,
left_bndry, "all_faces",
flux(dcoll, lbnd_tpair))
# right boundary weak-flux terms
+ op.project(dcoll,
right_bndry, "all_faces",
flux(dcoll, rbnd_tpair))
# interior weak-flux terms
+ op.project(dcoll,
FACE_RESTR_INTERIOR, "all_faces",
flux(dcoll, interior_tpair)))
def weak_grad(self, *args):
return op.weak_local_grad(self, *args)
def test_gradient(actx_factory,
form,
dim,
order,
vectorize,
nested,
visualize=False):
actx = actx_factory()
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for n in [4, 6, 8]:
mesh = mgen.generate_regular_rect_mesh(a=(-1, ) * dim,
b=(1, ) * dim,
nelements_per_axis=(n, ) * dim)
dcoll = DiscretizationCollection(actx, mesh, order=order)
def f(x):
result = dcoll.zeros(actx) + 1
for i in range(dim - 1):
result = result * actx.np.sin(np.pi * x[i])
result = result * actx.np.cos(np.pi / 2 * x[dim - 1])
return result
def grad_f(x):
result = make_obj_array(
[dcoll.zeros(actx) + 1 for _ in range(dim)])
for i in range(dim - 1):
for j in range(i):
result[i] = result[i] * actx.np.sin(np.pi * x[j])
result[i] = result[i] * np.pi * actx.np.cos(np.pi * x[i])
for j in range(i + 1, dim - 1):
result[i] = result[i] * actx.np.sin(np.pi * x[j])
result[i] = result[i] * actx.np.cos(np.pi / 2 * x[dim - 1])
for j in range(dim - 1):
result[dim - 1] = result[dim - 1] * actx.np.sin(np.pi * x[j])
result[dim -
1] = result[dim - 1] * (-np.pi / 2 *
actx.np.sin(np.pi / 2 * x[dim - 1]))
return result
x = thaw(dcoll.nodes(), actx)
if vectorize:
u = make_obj_array([(i + 1) * f(x) for i in range(dim)])
else:
u = f(x)
def get_flux(u_tpair):
dd = u_tpair.dd
dd_allfaces = dd.with_dtag("all_faces")
normal = thaw(dcoll.normal(dd), actx)
u_avg = u_tpair.avg
if vectorize:
if nested:
flux = make_obj_array(
[u_avg_i * normal for u_avg_i in u_avg])
else:
flux = np.outer(u_avg, normal)
else:
flux = u_avg * normal
return op.project(dcoll, dd, dd_allfaces, flux)
dd_allfaces = DOFDesc("all_faces")
if form == "strong":
grad_u = (
op.local_grad(dcoll, u, nested=nested)
# No flux terms because u doesn't have inter-el jumps
)
elif form == "weak":
grad_u = op.inverse_mass(
dcoll,
-op.weak_local_grad(dcoll, u, nested=nested) # pylint: disable=E1130
+ # noqa: W504
op.face_mass(
dcoll,
dd_allfaces,
# Note: no boundary flux terms here because u_ext == u_int == 0
sum(
get_flux(utpair)
for utpair in op.interior_trace_pairs(dcoll, u))))
else:
raise ValueError("Invalid form argument.")
if vectorize:
expected_grad_u = make_obj_array([(i + 1) * grad_f(x)
for i in range(dim)])
if not nested:
expected_grad_u = np.stack(expected_grad_u, axis=0)
else:
expected_grad_u = grad_f(x)
if visualize:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(dcoll,
vis_order=order if dim == 3 else dim + 3)
filename = (
f"test_gradient_{form}_{dim}_{order}"
f"{'_vec' if vectorize else ''}{'_nested' if nested else ''}.vtu"
)
vis.write_vtk_file(filename, [
("u", u),
("grad_u", grad_u),
("expected_grad_u", expected_grad_u),
],
overwrite=True)
rel_linf_err = actx.to_numpy(
op.norm(dcoll, grad_u - expected_grad_u, np.inf) /
op.norm(dcoll, expected_grad_u, np.inf))
eoc_rec.add_data_point(1. / n, rel_linf_err)
print("L^inf error:")
print(eoc_rec)
assert (eoc_rec.order_estimate() >= order - 0.5
or eoc_rec.max_error() < 1e-11)