import numpy as np
[docs]
def get_interface_states(grid, dt,
u, v,
ldelta_ux, ldelta_vx,
ldelta_uy, ldelta_vy):
r"""
Construct the interface states for the burgers equation:
.. math::
u_t + u u_x + v u_y = 0
v_t + u v_x + v v_y = 0
Compute the unsplit predictions of u and v on both the x- and
y-interfaces.
Parameters
----------
grid : Grid2d
The grid object
dt : float
The timestep
u, v : ndarray
x-velocity and y-velocity
ldelta_ux, ldelta_uy: ndarray
Limited slopes of the x-velocity in the x and y directions
ldelta_vx, ldelta_vy: ndarray
Limited slopes of the y-velocity in the x and y directions
Returns
-------
out : ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray
get the predictions of the left and right states of u and v on both the x- and
y-interfaces interface states without the transverse terms.
"""
u_xl = grid.scratch_array()
u_xr = grid.scratch_array()
u_yl = grid.scratch_array()
u_yr = grid.scratch_array()
v_xl = grid.scratch_array()
v_xr = grid.scratch_array()
v_yl = grid.scratch_array()
v_yr = grid.scratch_array()
# first predict u and v to both interfaces, considering only the normal
# part of the predictor. These are the 'hat' states.
dtdx = dt / grid.dx
dtdy = dt / grid.dy
# u on x-edges
u_xl.ip(1, buf=2)[:, :] = u.v(buf=2) + \
0.5 * (1.0 - dtdx * u.v(buf=2)) * ldelta_ux.v(buf=2)
u_xr.v(buf=2)[:, :] = u.v(buf=2) - \
0.5 * (1.0 + dtdx * u.v(buf=2)) * ldelta_ux.v(buf=2)
# v on x-edges
v_xl.ip(1, buf=2)[:, :] = v.v(buf=2) + \
0.5 * (1.0 - dtdx * u.v(buf=2)) * ldelta_vx.v(buf=2)
v_xr.v(buf=2)[:, :] = v.v(buf=2) - \
0.5 * (1.0 + dtdx * u.v(buf=2)) * ldelta_vx.v(buf=2)
# u on y-edges
u_yl.jp(1, buf=2)[:, :] = u.v(buf=2) + \
0.5 * (1.0 - dtdy * v.v(buf=2)) * ldelta_uy.v(buf=2)
u_yr.v(buf=2)[:, :] = u.v(buf=2) - \
0.5 * (1.0 + dtdy * v.v(buf=2)) * ldelta_uy.v(buf=2)
# v on y-edges
v_yl.jp(1, buf=2)[:, :] = v.v(buf=2) + \
0.5 * (1.0 - dtdy * v.v(buf=2)) * ldelta_vy.v(buf=2)
v_yr.v(buf=2)[:, :] = v.v(buf=2) - \
0.5 * (1.0 + dtdy * v.v(buf=2)) * ldelta_vy.v(buf=2)
return u_xl, u_xr, u_yl, u_yr, v_xl, v_xr, v_yl, v_yr
[docs]
def apply_transverse_corrections(grid, dt,
u_xl, u_xr,
u_yl, u_yr,
v_xl, v_xr,
v_yl, v_yr):
r"""
Parameters
----------
grid : Grid2d
The grid object
dt : float
The timestep
u_xl, u_xr : ndarray ndarray
left and right states of x-velocity in x-interface.
u_yl, u_yr : ndarray ndarray
left and right states of x-velocity in y-interface.
v_xl, v_xr : ndarray ndarray
left and right states of y-velocity in x-interface.
v_yl, u_yr : ndarray ndarray
left and right states of y-velocity in y-interface.
Returns
-------
out : ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray
correct the interface states of the left and right states of u and v on both the x- and
y-interfaces interface states with the transverse terms.
"""
dtdx = dt / grid.dx
dtdy = dt / grid.dy
# now get the normal advective velocities on the interfaces by solving
# the Riemann problem.
uhat_adv = riemann(grid, u_xl, u_xr)
vhat_adv = riemann(grid, v_yl, v_yr)
# now that we have the advective velocities, upwind the left and right
# states using the appropriate advective velocity.
# on the x-interfaces, we upwind based on uhat_adv
u_xint = upwind(grid, u_xl, u_xr, uhat_adv)
v_xint = upwind(grid, v_xl, v_xr, uhat_adv)
# on the y-interfaces, we upwind based on vhat_adv
u_yint = upwind(grid, u_yl, u_yr, vhat_adv)
v_yint = upwind(grid, v_yl, v_yr, vhat_adv)
# at this point, these states are the `hat' states -- they only
# considered the normal to the interface portion of the predictor.
ubar = grid.scratch_array()
vbar = grid.scratch_array()
ubar.v(buf=2)[:, :] = 0.5 * (uhat_adv.v(buf=2) + uhat_adv.ip(1, buf=2))
vbar.v(buf=2)[:, :] = 0.5 * (vhat_adv.v(buf=2) + vhat_adv.jp(1, buf=2))
# the transverse term for the u states on x-interfaces
u_xl.ip(1, buf=2)[:, :] += -0.5 * dtdy * vbar.v(buf=2) * \
(u_yint.jp(1, buf=2) - u_yint.v(buf=2))
u_xr.v(buf=2)[:, :] += -0.5 * dtdy * vbar.v(buf=2) * \
(u_yint.jp(1, buf=2) - u_yint.v(buf=2))
# the transverse term for the v states on x-interfaces
v_xl.ip(1, buf=2)[:, :] += -0.5 * dtdy * vbar.v(buf=2) * \
(v_yint.jp(1, buf=2) - v_yint.v(buf=2))
v_xr.v(buf=2)[:, :] += -0.5 * dtdy * vbar.v(buf=2) * \
(v_yint.jp(1, buf=2) - v_yint.v(buf=2))
# the transverse term for the v states on y-interfaces
v_yl.jp(1, buf=2)[:, :] += -0.5 * dtdx * ubar.v(buf=2) * \
(v_xint.ip(1, buf=2) - v_xint.v(buf=2))
v_yr.v(buf=2)[:, :] += -0.5 * dtdx * ubar.v(buf=2) * \
(v_xint.ip(1, buf=2) - v_xint.v(buf=2))
# the transverse term for the u states on y-interfaces
u_yl.jp(1, buf=2)[:, :] += -0.5 * dtdx * ubar.v(buf=2) * \
(u_xint.ip(1, buf=2) - u_xint.v(buf=2))
u_yr.v(buf=2)[:, :] += -0.5 * dtdx * ubar.v(buf=2) * \
(u_xint.ip(1, buf=2) - u_xint.v(buf=2))
return u_xl, u_xr, u_yl, u_yr, v_xl, v_xr, v_yl, v_yr
[docs]
def construct_unsplit_fluxes(grid,
u_xl, u_xr,
u_yl, u_yr,
v_xl, v_xr,
v_yl, v_yr):
r"""
Construct the interface fluxes for the burgers equation:
.. math::
u_t + u u_x + v u_y = 0
v_t + u v_x + v v_y = 0
Parameters
----------
grid : Grid2d
The grid object
u_xl, u_xr : ndarray ndarray
left and right states of x-velocity in x-interface.
u_yl, u_yr : ndarray ndarray
left and right states of x-velocity in y-interface.
v_xl, v_xr : ndarray ndarray
left and right states of y-velocity in x-interface.
v_yl, u_yr : ndarray ndarray
left and right states of y-velocity in y-interface.
-------
Returns
-------
out : ndarray, ndarray, ndarray, ndarray
The u,v fluxes on the x- and y-interfaces
"""
# Solve for riemann problem for the second time
# Get corrected normal advection velocity (MAC)
u_MAC = riemann_and_upwind(grid, u_xl, u_xr)
v_MAC = riemann_and_upwind(grid, v_yl, v_yr)
# Upwind using the transverse corrected normal advective velocity
ux = upwind(grid, u_xl, u_xr, u_MAC)
vx = upwind(grid, v_xl, v_xr, u_MAC)
uy = upwind(grid, u_yl, u_yr, v_MAC)
vy = upwind(grid, v_yl, v_yr, v_MAC)
# construct the flux
fu_x = grid.scratch_array()
fv_x = grid.scratch_array()
fu_y = grid.scratch_array()
fv_y = grid.scratch_array()
fu_x.v(buf=2)[:, :] = 0.5 * ux.v(buf=2) * u_MAC.v(buf=2)
fv_x.v(buf=2)[:, :] = 0.5 * vx.v(buf=2) * u_MAC.v(buf=2)
fu_y.v(buf=2)[:, :] = 0.5 * uy.v(buf=2) * v_MAC.v(buf=2)
fv_y.v(buf=2)[:, :] = 0.5 * vy.v(buf=2) * v_MAC.v(buf=2)
return fu_x, fu_y, fv_x, fv_y
[docs]
def upwind(grid, q_l, q_r, s):
r"""
upwind the left and right states based on the specified input
velocity, s. The resulting interface state is q_int
Parameters
----------
grid : Grid2d
The grid object
q_l, q_r : ndarray
left and right states
s : ndarray
velocity
Returns
-------
out : ndarray
Upwinded state
"""
q_int = grid.scratch_array()
q_int.v(buf=2)[:, :] = np.where(s.v(buf=2) == 0.0,
0.5 * (q_l.v(buf=2) + q_r.v(buf=2)),
np.where(s.v(buf=2) > 0.0, q_l.v(buf=2), q_r.v(buf=2)))
return q_int
[docs]
def riemann(grid, q_l, q_r):
"""
Solve the Burger's Riemann problem given the input left and right
states and return the state on the interface.
This uses the expressions from Almgren, Bell, and Szymczak 1996.
Parameters
----------
grid : Grid2d
The grid object
q_l, q_r : ndarray
left and right states
Returns
-------
out : ndarray
Interface state
"""
s = grid.scratch_array()
s.v(buf=2)[:, :] = np.where(np.logical_and(q_l.v(buf=2) <= 0.0,
q_r.v(buf=2) >= 0.0),
0.0,
np.where(np.logical_and(q_l.v(buf=2) > 0.0,
q_l.v(buf=2) + q_r.v(buf=2) > 0.0),
q_l.v(buf=2), q_r.v(buf=2)))
return s
[docs]
def riemann_and_upwind(grid, q_l, q_r):
r"""
First solve the Riemann problem given q_l and q_r to give the
velocity on the interface and: use this velocity to upwind to
determine the state (q_l, q_r, or a mix) on the interface).
This differs from upwind, above, in that we don't take in a
velocity to upwind with).
Parameters
----------
grid : Grid2d
The grid object
q_l, q_r : ndarray
left and right states
Returns
-------
out : ndarray
Upwinded state
"""
s = riemann(grid, q_l, q_r)
return upwind(grid, q_l, q_r, s)
