def div_operator(discr, dd_vol, dd_faces, v, flux):
r"""Compute a DG divergence of vector-valued function *v* 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.
v: numpy.ndarray
obj array of :class:`~meshmode.dof_array.DOFArray` (or container of such)
representing the vector-valued functions for which divergence is to be
calculated
flux: numpy.ndarray
the boundary flux for each function in v
Returns
-------
meshmode.dof_array.DOFArray or numpy.ndarray
the dg divergence operator applied to vector-valued function(s) *v*.
"""
return -discr.inverse_mass(
op.weak_local_div(discr, dd_vol, v) -
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 div_operator(discr, u, flux):
r"""Compute a DG divergence for *u* with element boundary flux given in *flux*.
Parameters
----------
discr: grudge.eager.EagerDGDiscretization
the discretization to use
u: np.ndarray
the vector-valued function for which divergence is to be calculated
flux: np.ndarray
the boundary fluxes across the faces of the element
Returns
-------
meshmode.dof_array.DOFArray or numpy.ndarray
the dg divergence operator applied to vector-valued function *u*.
"""
from grudge.op import weak_local_div
return -discr.inverse_mass(weak_local_div(discr, u)
- 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 weak_div(self, *args):
return op.weak_local_div(self, *args)
def test_divergence(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 = make_obj_array(
[dcoll.zeros(actx) + (i + 1) for i in range(dim)])
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 div_f(x):
result = dcoll.zeros(actx)
for i in range(dim - 1):
deriv = dcoll.zeros(actx) + (i + 1)
for j in range(i):
deriv = deriv * actx.np.sin(np.pi * x[j])
deriv = deriv * np.pi * actx.np.cos(np.pi * x[i])
for j in range(i + 1, dim - 1):
deriv = deriv * actx.np.sin(np.pi * x[j])
deriv = deriv * actx.np.cos(np.pi / 2 * x[dim - 1])
result = result + deriv
deriv = dcoll.zeros(actx) + dim
for j in range(dim - 1):
deriv = deriv * actx.np.sin(np.pi * x[j])
deriv = deriv * (-np.pi / 2 * actx.np.sin(np.pi / 2 * x[dim - 1]))
result = result + deriv
return result
x = thaw(dcoll.nodes(), actx)
if vectorize:
u = make_obj_array([(i + 1) * f(x) for i in range(dim)])
if not nested:
u = np.stack(u, axis=0)
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)
flux = u_tpair.avg @ normal
return op.project(dcoll, dd, dd_allfaces, flux)
dd_allfaces = DOFDesc("all_faces")
if form == "strong":
div_u = (
op.local_div(dcoll, u)
# No flux terms because u doesn't have inter-el jumps
)
elif form == "weak":
div_u = op.inverse_mass(
dcoll,
-op.weak_local_div(dcoll, u) + # 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_div_u = make_obj_array([(i + 1) * div_f(x)
for i in range(dim)])
else:
expected_div_u = div_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_divergence_{form}_{dim}_{order}"
f"{'_vec' if vectorize else ''}{'_nested' if nested else ''}.vtu"
)
vis.write_vtk_file(filename, [
("u", u),
("div_u", div_u),
("expected_div_u", expected_div_u),
],
overwrite=True)
rel_linf_err = actx.to_numpy(
op.norm(dcoll, div_u - expected_div_u, np.inf) /
op.norm(dcoll, expected_div_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)