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# Solves the PDE:
# \begin{cases}
# u_t + 2\pi u_x = 0, \\
# u(0, t) = -\sin(2\pi t), \\
# u(x, 0) = \sin(x),
# \end{cases}
# on the domain $x \in [0, 2\pi]$. We closely follow Chapter 3 of
# "Nodal Discontinuous Galerkin Methods" by Hesthaven & Warburton.
# BEGINEXAMPLE
import numpy as np
import pyopencl as cl
from grudge.discretization import DiscretizationCollection
import grudge.op as op
from meshmode.mesh.generation import generate_box_mesh
from meshmode.array_context import PyOpenCLArrayContext
from grudge.dof_desc import BoundaryDomainTag, FACE_RESTR_INTERIOR
ctx = cl.create_some_context()
queue = cl.CommandQueue(ctx)
actx = PyOpenCLArrayContext(queue)
nel = 10
coords = np.linspace(0, 2*np.pi, nel)
mesh = generate_box_mesh((coords,),
boundary_tag_to_face={"left": ["-x"],
"right": ["+x"]})
dcoll = DiscretizationCollection(actx, mesh, order=1)
def initial_condition(x):
# 'x' contains ndim arrays.
# 'x[0]' gets the first coordinate value of all the nodes
return actx.np.sin(x[0])
def left_boundary_condition(x, t):
return actx.np.sin(x[0] - 2 * np.pi * t)
def flux(dcoll, u_tpair):
dd = u_tpair.dd
velocity = np.array([2 * np.pi])
normal = actx.thaw(dcoll.normal(dd))
v_dot_n = np.dot(velocity, normal)
u_upwind = actx.np.where(v_dot_n > 0,
u_tpair.int, u_tpair.ext)
return u_upwind * v_dot_n
vol_discr = dcoll.discr_from_dd("vol")
left_bndry = BoundaryDomainTag("left")
right_bndry = BoundaryDomainTag("right")
x_vol = actx.thaw(dcoll.nodes())
x_bndry = actx.thaw(dcoll.discr_from_dd(left_bndry).nodes())
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uh = initial_condition(x_vol)
dt = 0.001
t = 0
t_final = 0.5
# timestepper loop
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)))
duh_by_dt = op.inverse_mass(dcoll,
np.dot([2 * np.pi], Su) - lift)
# forward euler time step
uh = uh + dt * duh_by_dt
t += dt
# ENDEXAMPLE
# Plot the solution:
def u_exact(x, t):
return actx.np.sin(x[0] - 2 * np.pi * t)
assert op.norm(dcoll,
uh - u_exact(x_vol, t_final),
p=2) <= 0.1
import matplotlib.pyplot as plt
from arraycontext import to_numpy
plt.plot(to_numpy(actx.np.ravel(x_vol[0][0]), actx),
to_numpy(actx.np.ravel(uh[0]), actx), label="Numerical")
plt.plot(to_numpy(actx.np.ravel(x_vol[0][0]), actx),
to_numpy(actx.np.ravel(u_exact(x_vol, t_final)[0]), actx), label="Exact")
plt.xlabel("$x$")
plt.ylabel("$u$")
plt.legend()
plt.show()