diff --git a/Basis_Function.py b/Basis_Function.py
index da720f3479947e091712f989a213c586fb241ec0..14315ad7341c96d5e5379955eaf542dc98972b76 100644
--- a/Basis_Function.py
+++ b/Basis_Function.py
@@ -1,11 +1,12 @@
# -*- coding: utf-8 -*-
-"""
+"""Module for polynomial basis.
+
@author: Laura C. Kühle
"""
+from functools import cache
import numpy as np
from sympy import Symbol, integrate
-from functools import cache
from projection_utils import calculate_approximate_solution
@@ -24,9 +25,16 @@ class Basis:
Array of basis.
wavelet : ndarray
Array of wavelet.
- inv_mass: ndarray
+ inv_mass : ndarray
Inverse mass matrix.
- basis_projection: ndarray
+ basis_projection : Tuple[ndarray, ndarray]
+ Two arrays containing integrals of all basis vector
+ combinations evaluated on the left and right cell boundary,
+ respectively.
+ wavelet_projection : Tuple[ndarray, ndarray]
+ Two arrays containing integrals of all basis vector/
+ wavelet vector combinations evaluated on the left and right cell
+ boundary, respectively.
Methods
@@ -35,8 +43,9 @@ class Basis:
Calculate cell averages for a given projection.
"""
+
def __init__(self, polynomial_degree):
- """Initializes Vector.
+ """Initialize Basis.
Parameters
----------
@@ -58,7 +67,7 @@ class Basis:
return self._build_basis_vector(x)
def _build_basis_vector(self, eval_point):
- """Constructs basis vector.
+ """Construct basis vector.
Parameters
----------
@@ -76,11 +85,11 @@ class Basis:
@property
@cache
def wavelet(self):
- """Return basis vector."""
+ """Return wavelet vector."""
return self._build_wavelet_vector(z)
def _build_wavelet_vector(self, eval_point):
- """Constructs wavelet vector.
+ """Construct wavelet vector.
Parameters
----------
@@ -102,7 +111,7 @@ class Basis:
return self._build_inverse_mass_matrix()
def _build_inverse_mass_matrix(self):
- """Constructs inverse mass matrix.
+ """Construct inverse mass matrix.
Returns
-------
@@ -192,8 +201,9 @@ class Basis:
class Legendre(Basis):
"""Class for Legendre basis."""
+
def _build_basis_vector(self, eval_point):
- """Constructs basis vector.
+ """Construct basis vector.
Parameters
----------
@@ -209,7 +219,7 @@ class Legendre(Basis):
return self._calculate_legendre_vector(eval_point)
def _calculate_legendre_vector(self, eval_point):
- """Constructs Legendre vector.
+ """Construct Legendre vector.
Parameters
----------
@@ -239,8 +249,9 @@ class Legendre(Basis):
class OrthonormalLegendre(Legendre):
"""Class for orthonormal Legendre basis."""
+
def _build_basis_vector(self, eval_point):
- """Constructs basis vector.
+ """Construct basis vector.
Parameters
----------
@@ -258,7 +269,7 @@ class OrthonormalLegendre(Legendre):
for degree in range(self._polynomial_degree+1)]
def _build_wavelet_vector(self, eval_point):
- """Constructs wavelet vector.
+ """Construct wavelet vector.
Parameters
----------
@@ -323,6 +334,14 @@ class OrthonormalLegendre(Legendre):
up to degree 4 for this application')
def _build_inverse_mass_matrix(self):
+ """Construct inverse mass matrix.
+
+ Returns
+ -------
+ ndarray
+ Inverse mass matrix.
+
+ """
mass_matrix = []
for i in range(self._polynomial_degree+1):
new_row = []
@@ -337,21 +356,26 @@ class OrthonormalLegendre(Legendre):
@property
@cache
def basis_projections(self):
- """Returns basis projection.
+ """Return basis projection.
+
+ Construct matrices containing the integrals of the
+ product of two basis vectors for every degree combination evaluated
+ at the left and right cell boundary.
Returns
-------
- ndarray
- Array containing the basis projection based on the integrals of
- the product of two basis vectors for each degree combination.
+ left_basis_projection : ndarray
+ Array containing the left basis projection.
+ right_basis_projection : ndarray
+ Array containing the right basis projection.
"""
- basis_projection_left = self._build_basis_matrix(z, 0.5 * (z - 1))
- basis_projection_right = self._build_basis_matrix(z, 0.5 * (z + 1))
- return basis_projection_left, basis_projection_right
+ left_basis_projection = self._build_basis_matrix(z, 0.5 * (z - 1))
+ right_basis_projection = self._build_basis_matrix(z, 0.5 * (z + 1))
+ return left_basis_projection, right_basis_projection
def _build_basis_matrix(self, first_param, second_param):
- """Constructs a basis matrix.
+ """Construct a basis matrix.
Parameters
----------
@@ -380,25 +404,29 @@ class OrthonormalLegendre(Legendre):
@property
@cache
def multiwavelet_projections(self):
- """Returns wavelet projection.
+ """Return wavelet projection.
+
+ Construct matrices containing the integrals of the
+ product of a basis vector and a wavelet vector for every degree
+ combination evaluated at the left and right cell boundary.
Returns
-------
- ndarray
- Array containing the multiwavelet projection based on the integrals
- of the product of a basis vector and a wavelet vector for each
- degree combination.
+ left_wavelet_projection : ndarray
+ Array containing the left multiwavelet projection.
+ right_wavelet_projection : ndarray
+ Array containing the right multiwavelet projection.
"""
- wavelet_projection_left = self._build_multiwavelet_matrix(
+ left_wavelet_projection = self._build_multiwavelet_matrix(
z, -0.5*(z-1), True)
- wavelet_projection_right = self._build_multiwavelet_matrix(
+ right_wavelet_projection = self._build_multiwavelet_matrix(
z, 0.5*(z+1), False)
- return wavelet_projection_left, wavelet_projection_right
+ return left_wavelet_projection, right_wavelet_projection
def _build_multiwavelet_matrix(self, first_param, second_param,
is_left_matrix):
- """Constructs a multiwavelet matrix.
+ """Construct a multiwavelet matrix.
Parameters
----------
@@ -439,7 +467,7 @@ class OrthonormalLegendre(Legendre):
Notes
-----
- To increase speed. this function uses a simplified calculation
+ To increase speed, this function uses a simplified calculation
specific to the orthonormal Legendre polynomial basis.
Parameters
@@ -459,7 +487,6 @@ class OrthonormalLegendre(Legendre):
projection.
"""
-
cell_averages = np.array([projection[0] / np.sqrt(2)])
if add_reconstructions:
@@ -478,11 +505,6 @@ class OrthonormalLegendre(Legendre):
def _calculate_reconstructions(self, projection):
"""Calculate left and right reconstructions for a given projection.
- Notes
- -----
- To increase speed. this function uses a simplified calculation
- specific to the orthonormal Legendre polynomial basis.
-
Parameters
----------
projection : ndarray
@@ -495,8 +517,12 @@ class OrthonormalLegendre(Legendre):
right_reconstruction : list
List containing right reconstructions for given projection.
- """
+ Notes
+ -----
+ To increase speed. this function uses a simplified calculation
+ specific to the orthonormal Legendre polynomial basis.
+ """
left_reconstructions = [
sum(projection[degree][cell] * (-1)**degree
* np.sqrt(degree + 0.5)