def sym_wave(dim, sym_phi):
"""Return symbolic expressions for the wave equation system given a desired
solution. (Note: In order to support manufactured solutions, we modify the wave
equation to add a source term (f). If the solution is exact, this term should
be 0.)
"""
sym_c = pmbl.var("c")
sym_coords = prim.make_sym_vector("x", dim)
sym_t = pmbl.var("t")
# f = phi_tt - c^2 * div(grad(phi))
sym_f = sym.diff(sym_t)(sym.diff(sym_t)(sym_phi)) - sym_c**2\
* sym.div(sym.grad(dim, sym_phi))
# u = phi_t
sym_u = sym.diff(sym_t)(sym_phi)
# v = c*grad(phi)
sym_v = [sym_c * sym.diff(sym_coords[i])(sym_phi) for i in range(dim)]
# rhs(u part) = c*div(v) + f
# rhs(v part) = c*grad(u)
sym_rhs = flat_obj_array(sym_c * sym.div(sym_v) + sym_f,
make_obj_array([sym_c]) * sym.grad(dim, sym_u))
return sym_u, sym_v, sym_f, sym_rhs
def test_symbolic_diff(sym_f, expected_sym_df):
"""
Compute the symbolic derivative of an expression and compare it to an
expected result.
"""
sym_df = sym.diff(pmbl.var("x"))(sym_f)
assert sym_df == expected_sym_df
def test_diffusion_accuracy(actx_factory,
problem,
nsteps,
dt,
scales,
order,
visualize=False):
"""
Checks the accuracy of the diffusion operator by solving the heat equation for a
given problem setup.
"""
actx = actx_factory()
p = problem
sym_diffusion_u = sym_diffusion(p.dim, p.sym_alpha, p.sym_u)
# In order to support manufactured solutions, we modify the heat equation
# to add a source term f. If the solution is exact, this term should be 0.
sym_t = pmbl.var("t")
sym_f = sym.diff(sym_t)(p.sym_u) - sym_diffusion_u
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for n in scales:
mesh = p.get_mesh(n)
from grudge.eager import EagerDGDiscretization
from meshmode.discretization.poly_element import \
QuadratureSimplexGroupFactory, \
PolynomialWarpAndBlendGroupFactory
discr = EagerDGDiscretization(
actx,
mesh,
discr_tag_to_group_factory={
DISCR_TAG_BASE: PolynomialWarpAndBlendGroupFactory(order),
DISCR_TAG_QUAD: QuadratureSimplexGroupFactory(3 * order),
})
nodes = thaw(actx, discr.nodes())
def sym_eval(expr, t):
return sym.EvaluationMapper({"x": nodes, "t": t})(expr)
alpha = sym_eval(p.sym_alpha, 0.)
if isinstance(alpha, DOFArray):
discr_tag = DISCR_TAG_QUAD
else:
discr_tag = DISCR_TAG_BASE
def get_rhs(t, u):
return (
diffusion_operator(discr,
quad_tag=discr_tag,
alpha=alpha,
boundaries=p.get_boundaries(discr, actx, t),
u=u) + sym_eval(sym_f, t))
t = 0.
u = sym_eval(p.sym_u, t)
from mirgecom.integrators import rk4_step
for _ in range(nsteps):
u = rk4_step(u, t, dt, get_rhs)
t += dt
expected_u = sym_eval(p.sym_u, t)
rel_linf_err = (discr.norm(u - expected_u, np.inf) /
discr.norm(expected_u, np.inf))
eoc_rec.add_data_point(1. / n, rel_linf_err)
if visualize:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(discr, discr.order + 3)
vis.write_vtk_file(
"diffusion_accuracy_{order}_{n}.vtu".format(order=order, n=n),
[
("u", u),
("expected_u", expected_u),
])
print("L^inf error:")
print(eoc_rec)
# Expected convergence rates from Hesthaven/Warburton book
expected_order = order + 1 if order % 2 == 0 else order
assert (eoc_rec.order_estimate() >= expected_order - 0.5
or eoc_rec.max_error() < 1e-11)
