The element `result` then corresponds to the lower-order (`n`) and the second element `result` corresponds to the higher-order (`2*n + 1`) approximation of the integral. A one-dimensional integral :math:`\\int_a^b f(x) dx` can be approximated by ``result = np.sum(w * f(x), axis=1)``. For the Gauss-Kronrod quadrature, `x` is a one dimensional numpy array of shape `(2*n+1,)` and `w` is a two-dimensional array of shape `(2, 2*n+1)`. A one-dimensional integral :math:`\\int_a^b f(x) dx` can be approximated by an order ``n`` quadrature via ``np.sum(w * f(x))``. Possible choices:\n * **gausslegendre**: order ``n`` Gauss-Legendre\n * **kronrod**: order ``n`` and ``2 n + 1`` Gauss-Kronrod\n * **trapezoidal**: ``n`` point trapezoidal rule\n * **trapezoidal-2**: ``n`` and ``2 n - 1`` point trapezoidal rule\n Returns - x : Numpy float array abscissa values w : Numpy float array quadrature weights Notes - For the Guass-Legendre quadrature, both `x` and `w` are one-dimensional numpy arrays of shape `(n, )`. Must be positive, n > 0 quadrature : string Quadrature rule to use. Parameters - a : float Lower limit of integration. def calc_abscissas_and_weights ( a, b, n, quadrature ): """Abscissas and weights for fixed-order integration quadratures. linspace ( - 1, 1, n ) dx = 2 / ( n - 1 ) if n = 2 : w = np. Returns - x : Numpy float array abscissa values w : Numpy float array quadrature weights """ if n = 2'. import line_segment, _common _all_ = def _trapezoidal_rule ( n ): """Calculate trapezoidal abcissas and weights on an interval Parameters - n : int Number of quadrature points. """Tools for calculating numeric integrals.""" import numpy as np from. A list of tkwant authors can be found in # the file AUTHORS.rst at the top-level directory of this distribution and at #.
#ABSCISSA QUADRATURE POINT LICENSE#
It is subject to the license terms in the file # LICENSE.rst found in the top-level directory of this distribution and at #. Thus the self-consistent way of getting the total flux is to add the fluxes of all the cells on a patch.# -*- coding: utf-8 -*- # Copyright 2016-2019 tkwant authors. That will force you to interpolate the flux to those locations, and then you face the question of how do you interpolate.įinally, when I think about it, since this is a finite volume method, the flux is constant across a cell (at least in orthogonal meshes). I don't think Gaussian quadratures are of much help here, as they require nodes at specific locations. Extract the points, and use a trapezoidal or a Romberg integration. I have never done this one, but search the forum for integrating fluxes over patches. I thought your region might be non-rectangular, and thus a 4x4 array of points would extend outside of it.Īs for your problem, I have two suggestions: How can I find a polynomial which can represent my data points? I want to set some points (say a 4 X 4 array) and find out the volume flow. What do you mean when you say some of it is identically zero? I am attaching here the geometry, the cut and the region where I have the computation results. However, I did not fully understand your explanation.
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