from pyro.multigrid import MG
[docs]
def get_lap(grid, a):
r"""
Parameters
----------
grid: grid object
a: ndarray
the variable that we want to find the laplacian of
Returns
-------
out : ndarray (laplacian of state a)
"""
lap = grid.scratch_array()
lap.v(buf=2)[:, :] = (a.ip(1, buf=2) - 2.0*a.v(buf=2) + a.ip(-1, buf=2)) / grid.dx**2 \
+ (a.jp(1, buf=2) - 2.0*a.v(buf=2) + a.jp(-1, buf=2)) / grid.dy**2
return lap
[docs]
def diffuse(my_data, rp, dt, scalar_name, A):
r"""
A routine to solve the Helmhotlz equation with constant coefficient
and update the state.
(a + b \lap) phi = f
using Crank-Nicolson discretization with multigrid V-cycle.
Parameters
----------
my_data : CellCenterData2d object
The data object containing the grid and advective scalar that
we are advecting.
rp : RuntimeParameters object
The runtime parameters for the simulation
dt : float
The timestep we are advancing through.
scalar_name : str
The name of the variable contained in my_data that we are
advecting
A: ndarray
The advective source term for diffusion
Returns
-------
out : ndarray (solution of the Helmholtz equation)
"""
myg = my_data.grid
a = my_data.get_var(scalar_name)
eps = rp.get_param("diffusion.eps")
# Create the multigrid with a constant diffusion coefficient
# the equation has form:
# (alpha - beta L) phi = f
#
# with alpha = 1
# beta = (dt/2) k
# f = phi + (dt/2) k L phi - A
#
# Same as Crank-Nicolson discretization except with an extra
# advection source term)
mg = MG.CellCenterMG2d(myg.nx, myg.ny,
xmin=myg.xmin, xmax=myg.xmax,
ymin=myg.ymin, ymax=myg.ymax,
xl_BC_type=my_data.BCs[scalar_name].xlb,
xr_BC_type=my_data.BCs[scalar_name].xrb,
yl_BC_type=my_data.BCs[scalar_name].ylb,
yr_BC_type=my_data.BCs[scalar_name].yrb,
alpha=1.0, beta=0.5*dt*eps,
verbose=0)
# Compute the RHS: f
f = mg.soln_grid.scratch_array()
lap = get_lap(myg, a)
f.v()[:, :] = a.v() + 0.5*dt*eps * lap.v() - dt*A.v()
mg.init_RHS(f)
# initial guess is zeros
mg.init_zeros()
mg.solve(rtol=1.e-12)
# perform the diffusion update
a.v()[:, :] = mg.get_solution().v()
[docs]
def apply_diffusion_corrections(grid, dt, eps,
u, v,
u_xl, u_xr,
u_yl, u_yr,
v_xl, v_xr,
v_yl, v_yr):
r"""
Apply diffusion correction term to the interface state
.. math::
u_t + u u_x + v u_y = eps (u_xx + u_yy)
v_t + u v_x + v v_y = eps (v_xx + v_yy)
We use a second-order (piecewise linear) unsplit Godunov method
(following Colella 1990).
Our convection is that the fluxes are going to be defined on the
left edge of the computational zones::
| | | |
| | | |
-+------+------+------+------+------+------+--
| i-1 | i | i+1 |
a_l,i a_r,i a_l,i+1
a_r,i and a_l,i+1 are computed using the information in
zone i,j.
Parameters
----------
grid : Grid2d
The grid object
dt : float
The timestep
eps : float
The viscosity
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
unsplit predictions of the left and right states of u and v on both the x- and
y-interfaces along with diffusion correction terms.
"""
#apply diffusion correction to the interface
lap_u = get_lap(grid, u)
lap_v = get_lap(grid, v)
u_xl.ip(1, buf=2)[:, :] += 0.5 * eps * dt * lap_u.v(buf=2)
u_yl.jp(1, buf=2)[:, :] += 0.5 * eps * dt * lap_u.v(buf=2)
u_xr.v(buf=2)[:, :] += 0.5 * eps * dt * lap_u.v(buf=2)
u_yr.v(buf=2)[:, :] += 0.5 * eps * dt * lap_u.v(buf=2)
v_xl.ip(1, buf=2)[:, :] += 0.5 * eps * dt * lap_v.v(buf=2)
v_yl.jp(1, buf=2)[:, :] += 0.5 * eps * dt * lap_v.v(buf=2)
v_xr.v(buf=2)[:, :] += 0.5 * eps * dt * lap_v.v(buf=2)
v_yr.v(buf=2)[:, :] += 0.5 * eps * dt * lap_v.v(buf=2)
return u_xl, u_xr, u_yl, u_yr, v_xl, v_xr, v_yl, v_yr
