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# -*- coding: utf-8 -*-
import numpy as np
from . import diff
from . import utils
def calc_christoffel(x, z, metric):
"""
Calculate Christoffel symbols
i 1 il / \
Γ = -γ | ∂ γ + ∂ γ - ∂ γ |
jk 2 \ j kl k jl l jk /
using finite differences.
:param array_like x: 1D array of x coordinates.
:param array_like z: 1D array of z-coordinates.
:param array_like metric: 4D array of spatial metric values at the grid
formed by x and z. metric[i, j, k, l] is the ijth
component of the metric at the point (X=x[l],
Z=z[k]).
:rtype: array_like, shape (3, 3, 3, z.shape[0], x.shape[0])
:return: Christoffel symbols, first axis is the upper index, following two
axes are the two lower indices, final two axes correspond to the z
and x grid points respectively.
"""
X, Z = np.meshgrid(x, z)
dmetric = np.zeros((3,) + metric.shape)
dmetric[0] = diff.fd4(metric, 3, x[1] - x[0])
dmetric[2] = diff.fd4(metric, 2, z[1] - z[0])
dmetric[1, 0, 0] = 0.0
dmetric[1, 1, 1] = 0.0
dmetric[1, 2, 2] = 0.0
dmetric[1, 0, 1] = np.where(np.abs(X) > 1e-8, (metric[0, 0] - metric[1, 1]) / X, dmetric[0, 0, 0] - dmetric[0, 1, 1])
dmetric[1, 1, 0] = dmetric[1, 0, 1]
dmetric[1, 0, 2] = 0.0
dmetric[1, 2, 0] = 0.0
dmetric[1, 1, 2] = np.where(np.abs(X) > 1e-8, metric[0, 2] / X, dmetric[0, 0, 2])
dmetric[1, 2, 1] = dmetric[1, 1, 2]
metric_u = utils.matrix_invert(metric)
Gamma = np.empty_like(dmetric)
for i in range(3):
for j in range(3):
for k in range(3):
Gamma[i, j, k] = 0.5 * np.einsum('k...,k...', metric_u[i], dmetric[j, k] + dmetric[k, j] - dmetric[:, k, j])
return Gamma
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