def _set_derivative_method(self):
if self.solver_type == 'spectral':
self.dx = lambda x: fourier(x, self.ikx)
self.dy = lambda x: fourier(x, self.iky)
else:
self.dx = lambda x: fd_x_4(x, self.ebdyc.grid.xh)
self.dy = lambda x: fd_y_4(x, self.ebdyc.grid.yh)
def laplacian(self, ff, derivative_type='spectral'):
"""
Compute the laplacian of a function defined on the embedded boundary collection
Inputs:
ff, EmbeddedFunction: function to take the gradient of
derivative_type: 'spectral' or 'fourth'
in both cases, spectral differentation is used in the radial
regions. On the grid, if 'spectral', then the function is cutoff
and fourier based estimation of the derivative is used;
if 'fourth', then the function is NOT cutoff and fourth-order
centered differences are used. This may provide better accuracy
when the cutoff windows are thin. However, if the cutoff windows
are not at least 2h, then this can give catastrophically bad values
Outputs:
lapf: EmbeddedFunction giving the laplacian of f
"""
f, _, fr_list = ff.get_components()
if derivative_type == 'spectral':
fc = f * self.grid_step
fc[self.ext] = 0.0
fch = np.fft.fft2(fc)
lapfh = fch * self.lap
lapf = np.fft.ifft2(lapfh).real
else:
fx = fd_x_4(f, self.grid.xh)
fy = fd_y_4(f, self.grid.yh)
fxx = fd_x_4(fx, self.grid.xh)
fyy = fd_y_4(fy, self.grid.yh)
# compute on the radial grid
lapfrs = []
for i in range(self.N):
lapfr = self.ebdys[i].laplacian(lapf, fr_list[i])
lapfrs.append(lapfr)
# set to 0 on regular grid in the exterior region
lapf *= self.phys
# generate EmbeddedFunctions
lapff = EmbeddedFunction(self)
lapff.load_data(lapf, lapfrs)
return lapff
def gradient(self, ff, derivative_type='spectral'):
"""
Compute the gradient of a function defined on the embedded boundary collection
Inputs:
ff, EmbeddedFunction: function to take the gradient of
derivative_type: 'spectral' or 'fourth'
in both cases, spectral differentation is used in the radial
regions. On the grid, if 'spectral', then the function is cutoff
and fourier based estimation of the derivative is used;
if 'fourth', then the function is NOT cutoff and fourth-order
centered differences are used. This may provide better accuracy
when the cutoff windows are thin. However, if the cutoff windows
are not at least 2h, then this can give catastrophically bad values
Outputs:
fx, fy: EmbeddedFunctions giving each required derivative
"""
f, _, fr_list = ff.get_components()
if derivative_type == 'spectral':
fc = f * self.grid_step
fc[self.ext] = 0.0
fch = np.fft.fft2(fc)
fxh = fch * self.ikx
fyh = fch * self.iky
fx = np.fft.ifft2(fxh).real
fy = np.fft.ifft2(fyh).real
else:
fx = fd_x_4(f, self.grid.xh)
fy = fd_y_4(f, self.grid.yh)
# compute on the radial grid
fxrs, fyrs = [], []
for i in range(self.N):
fxr, fyr = self[i].gradient(fx, fy, fr_list[i])
fxrs.append(fxr)
fyrs.append(fyr)
# set to 0 on regular grid in the exterior region
fx *= self.phys
fy *= self.phys
# generate EmbeddedFunctions
ffx = EmbeddedFunction(self)
ffy = EmbeddedFunction(self)
ffx.load_data(fx, fxrs)
ffy.load_data(fy, fyrs)
return ffx, ffy
by = ebdy.bdy.y + dt * vb
bx, by, new_t = arc_length_parameterize(bx, by, return_t=True)
bu_interp = interp1d(0, 2 * np.pi, bdy.dt, ub, p=True)
bv_interp = interp1d(0, 2 * np.pi, bdy.dt, vb, p=True)
# old boundary velocity values have to be in the correct place
ubo_new_parm = bu_interp(new_t)
vbo_new_parm = bv_interp(new_t)
ubo = ub.copy()
vbo = vb.copy()
x_tracers.append(bx)
y_tracers.append(by)
# take gradients of the velocity fields
dx = lambda f: fd_x_4(f, grid.xh, periodic_fix=True)
dy = lambda f: fd_y_4(f, grid.yh, periodic_fix=True)
if u_differentiation_type == 'spectral':
ux, uy = ebdyc.gradient2(u, xder, yder, cutoff=True)
vx, vy = ebdyc.gradient2(v, xder, yder, cutoff=True)
elif u_differentiation_type == 'fourth':
ux, uy = ebdyc.gradient2(u, dx, dy, cutoff=False)
vx, vy = ebdyc.gradient2(v, dx, dy, cutoff=False)
else:
ux = EmbeddedFunction(ebdyc)
uy = EmbeddedFunction(ebdyc)
vx = EmbeddedFunction(ebdyc)
vy = EmbeddedFunction(ebdyc)
ux.define_via_function(lambda x, y: ux_function(x, y, t))
uy.define_via_function(lambda x, y: uy_function(x, y, t))
vx.define_via_function(lambda x, y: vx_function(x, y, t))
err = max(errs)
print('Error in grid --> interface interpolation: {:0.2e}'.format(err))
# Interpolation of a function from radial to grid
frs = [test_func(ebdy.radial_x, ebdy.radial_y) for ebdy in ebdys]
fts = ebdyc.interpolate_radial_to_grid(frs)
fes = [test_func(ebdy.grid_ia_x, ebdy.grid_ia_y) for ebdy in ebdys]
errs = [np.abs(fe - ft).max() for fe, ft in zip(fes, fts)]
err = max(errs)
print('Error in radial --> grid interpolation: {:0.2e}'.format(err))
################################################################################
# Test derivatives
# fourth order accurate gradient on whole domain
dx = lambda x: fd_x_4(x, grid.xh, periodic_fix=True)
dy = lambda x: fd_y_4(x, grid.yh, periodic_fix=True)
fxe, fye, fxres, fyres = ebdyc.gradient(f, frs, dx, dy, cutoff=False)
fxt = test_func_x(grid.xg, grid.yg)
fyt = test_func_y(grid.xg, grid.yg)
err_x = np.abs(fxt - fxe)[ebdyc.phys].max()
err_y = np.abs(fyt - fye)[ebdyc.phys].max()
err = max(err_x, err_y)
print('Error in gradient, 4th order FD: {:0.2e}'.format(err))
# spectrally accurate gradient on whole domain
kxv = np.fft.fftfreq(grid.Nx, grid.xh / (2 * np.pi))
kyv = np.fft.fftfreq(grid.Ny, grid.yh / (2 * np.pi))
kx, ky = np.meshgrid(kxv, kyv, indexing='ij')
ikx, iky = 1j * kx, 1j * ky
dx = lambda x: fourier(x, ikx)
print('Error in radial --> grid interpolation: {:0.2e}'.format(err))
################################################################################
# Test derivatives
# radial gradient
frxe, frye = ebdy.radial_grid_derivatives(fr)
frxt = test_func_x(ebdy.radial_x, ebdy.radial_y)
fryt = test_func_y(ebdy.radial_x, ebdy.radial_y)
err_x = np.abs(frxt-frxe).max()
err_y = np.abs(fryt-frye).max()
err = max(err_x, err_y)
print('Error in radial grid differentiation: {:0.2e}'.format(err))
# fourth order accurate gradient on whole domain
dx = lambda x: fd_x_4(x, grid.xh, periodic_fix=not interior)
dy = lambda x: fd_y_4(x, grid.yh, periodic_fix=not interior)
fxe, fye, fxre, fyre = ebdy.gradient(f, fr, dx, dy)
fxt = test_func_x(grid.xg, grid.yg)
fyt = test_func_y(grid.xg, grid.yg)
err_x = np.abs(fxt-fxe)[ebdy.phys].max()
err_y = np.abs(fyt-fye)[ebdy.phys].max()
err = max(err_x, err_y)
print('Error in gradient, 4th order FD: {:0.2e}'.format(err))
# spectrally accurate gradient on whole domain
kxv = np.fft.fftfreq(grid.Nx, grid.xh/(2*np.pi))
kyv = np.fft.fftfreq(grid.Ny, grid.yh/(2*np.pi))
kx, ky = np.meshgrid(kxv, kyv, indexing='ij')
ikx, iky = 1j*kx, 1j*ky
dx = lambda x: fourier(x, ikx)
ebdyc.register_grid(grid)
# initial c field
c0 = EmbeddedFunction(ebdyc)
c0.define_via_function(c0_function)
def xder(f):
return np.fft.ifft2(np.fft.fft2(f) * ikx).real
def yder(f):
return np.fft.ifft2(np.fft.fft2(f) * iky).real
xder = lambda f: fd_x_4(f, grid.xh)
yder = lambda f: fd_y_4(f, grid.yh)
# now timestep
c = c0.copy()
t = 0.0
x_tracers = []
y_tracers = []
ts = []
new_ebdyc = ebdyc
qfs0 = QFS_Evaluator(new_ebdyc.ebdys[0].bdy_qfs,
True, [
Singular_SLP0,