import mesh.reconstruction as reconstruction
import numpy as np
[docs]def fvs(q, order, u, alpha):
"""
Perform Flux-Vector-Split (LF) finite differencing using WENO in 1d.
Parameters
----------
q : np array
input data with at least order+1 ghost zones
order : int
WENO order (k)
u : float
Advection velocity in this direction
alpha : float
Maximum characteristic speed
Returns
-------
f : np array
flux
"""
flux = u * q
flux_p = (flux + alpha * q) / 2
flux_m = (flux - alpha * q) / 2
flux_p_r = np.zeros_like(flux_p)
flux_m_l = np.zeros_like(flux_m)
Npoints = len(q)
for i in range(order, Npoints-order):
flux_p_r[i] = reconstruction.weno_upwind(flux_p[i-order:i+order-1],
order)
flux_m_l[i] = reconstruction.weno_upwind(flux_m[i+order-1:i-order:-1],
order)
flux[1:-1] = flux_p_r[1:-1] + flux_m_l[1:-1]
return flux
[docs]def fluxes(my_data, rp, dt):
r"""
Construct the fluxes through the interfaces for the linear advection
equation
.. math::
a_t + u a_x + v a_y = 0
We use a high-order flux split WENO method to construct the interface
fluxes. No Riemann problems are solved. The Lax-Friedrichs flux split
will probably make it diffusive; the lack of a transverse solver also
will not help.
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
Returns
-------
out : ndarray, ndarray
The fluxes on the x- and y-interfaces
"""
myg = my_data.grid
a = my_data.get_var("density")
# get the advection velocities
u = rp.get_param("advection.u")
v = rp.get_param("advection.v")
qx = myg.qx
qy = myg.qy
# --------------------------------------------------------------------------
# WENO fvs
# --------------------------------------------------------------------------
weno_order = rp.get_param("advection.weno_order")
assert(weno_order in (2, 3)), "Currently only implemented weno_order=2, 3"
assert(myg.ng > weno_order), "Need more ghosts than the weno_order"
q = a.v(buf=myg.ng)[:, :]
F_x = myg.scratch_array()
F_y = myg.scratch_array()
alpha = np.sqrt(u**2 + v**2)
# x-direction
for j in range(qy):
F_x.v(buf=myg.ng)[1:-1, j] = fvs(q[:, j], weno_order, u, alpha)[1:-1]
# y-direction
for i in range(qx):
F_y.v(buf=myg.ng)[i, 1:-1] = fvs(q[i, :], weno_order, v, alpha)[1:-1]
return F_x, F_y