def calculate_omega(
u,
optimal_linear_weights=OPTIMAL_SMOOTH_WEIGHTS,
epsilon=1e-6,
p=2,
):
"""Calculate linear weights for the three polynomial reconstructions."""
# see Equation (6) in ref [1]
indicator_kj = calculate_smoothness_indicators(u)
# p=2 is used by ref. [2]
alpha_kj = (np.array(optimal_linear_weights)[:, np.newaxis] /
(epsilon + indicator_kj)**p)
omega_kj = alpha_kj / duckarray.sum(alpha_kj, axis=-2, keepdims=True)
return omega_kj
def __call__(self, t: float) -> np.ndarray:
spatial_phase = (2 * np.pi * self.k * self.grid.reference_x
/ self.grid.period)
signals = duckarray.sin(self.omega * t + spatial_phase + self.phi)
reference_forcing = duckarray.sum(self.a * signals, axis=0)
return self.grid.resample(reference_forcing)
def reconstruct_left(u):
"""Reconstruct u at +1/2 cells with a left-biased stencil."""
coefficients = left_coefficients(u)
u_all = duckarray.stack(
[duckarray.roll(u, i, axis=-1) for i in [2, 1, 0, -1, -2]], axis=-1)
return duckarray.sum(coefficients * u_all, axis=-1)