Line data    Source code

       1            1 : // Distributed under the MIT License.
       2              : // See LICENSE.txt for details.
       3              : 
       4              : /// \file
       5              : /// Defines function definite_integral.
       6              : 
       7              : #pragma once
       8              : 
       9              : #include <cstddef>
      10              : 
      11              : /// \cond
      12              : class DataVector;
      13              : template <size_t>
      14              : class Mesh;
      15              : /// \endcond
      16              : 
      17              : /*!
      18              :  * \ingroup NumericalAlgorithmsGroup
      19              :  * \brief Compute the definite integral of a function over a manifold.
      20              :  *
      21              :  * Given a function \f$f\f$, compute its integral \f$I\f$ with respect to
      22              :  * the logical coordinates \f$\boldsymbol{\xi} = (\xi, \eta, \zeta)\f$.
      23              :  * E.g., in 1 dimension, \f$I = \int_{-1}^1 f d\xi\f$.
      24              :  *
      25              :  * The integral w.r.t. a different set of coordinates
      26              :  * \f$\boldsymbol{x} = \boldsymbol{x}(\boldsymbol{\xi})\f$ can be computed
      27              :  * by pre-multiplying \f$f\f$ by the Jacobian determinant
      28              :  * \f$J = \det d\boldsymbol{x}/d\boldsymbol{\xi}\f$ of the mapping
      29              :  * \f$\boldsymbol{x}(\boldsymbol{\xi})\f$. Note that, in the
      30              :  * \f$\boldsymbol{x}\f$ coordinates, the domain of integration is the image of
      31              :  * the logical cube (square in 2D, interval in 1D) under the mapping.
      32              :  *
      33              :  * The integral is computed by quadrature, using the quadrature rule for the
      34              :  * basis associated with the collocation points.
      35              :  *
      36              :  * If the mesh uses a Cartoon basis, the passed integrand must already be
      37              :  * multiplied by the appropriate Jacobian determinants (i.e. both the
      38              :  * $\boldsymbol{x}(\boldsymbol{\xi})$ and the cartesian to spherical/polar
      39              :  * coordinates mappings, the latter being either $x^2$ or $x$, respectively). If
      40              :  * the Mesh uses a Zernike basis, the passed integrand must already be
      41              :  * multiplied by the appropriate cartesian to spherical/polar Jacobain
      42              :  * determinant ($r$ for ZernikeB2 and $r^2$ for ZernikeB3).
      43              :  *
      44              :  * \param integrand the function to integrate.
      45              :  * \param mesh the Mesh defining the grid points on the manifold.
      46              :  */
      47              : template <size_t Dim>
      48            1 : double definite_integral(const DataVector& integrand, const Mesh<Dim>& mesh);