Line data    Source code

       1            0 : // Distributed under the MIT License.
       2              : // See LICENSE.txt for details.
       3              : 
       4              : #pragma once
       5              : 
       6              : #include <cstddef>
       7              : 
       8              : /// \cond
       9              : class DataVector;
      10              : template <size_t>
      11              : class Mesh;
      12              : namespace Spectral {
      13              : enum class Basis : uint8_t;
      14              : enum class Quadrature : uint8_t;
      15              : }  // namespace Spectral
      16              : /// \endcond
      17              : 
      18              : namespace Spectral {
      19              : /*!
      20              :  * \brief Weights to compute definite integrals.
      21              :  *
      22              :  * \details These are the coefficients to contract with the nodal
      23              :  * function values \f$f_k\f$ to approximate the definite integral \f$I[f]=\int
      24              :  * f(x)\mathrm{d}x\f$.
      25              :  *
      26              :  * Note that the term _quadrature_ also often refers to the quantity
      27              :  * \f$Q[f]=\int f(x)w(x)\mathrm{d}x\approx \sum_k f_k w_k\f$. Here, \f$w(x)\f$
      28              :  * denotes the basis-specific weight function w.r.t. to which the basis
      29              :  * functions \f$\Phi_k\f$ are orthogonal, i.e \f$\int\Phi_i(x)\Phi_j(x)w(x)=0\f$
      30              :  * for \f$i\neq j\f$. The weights \f$w_k\f$ approximate this inner product. To
      31              :  * approximate the definite integral \f$I[f]\f$ we must employ the
      32              :  * coefficients \f$\frac{w_k}{w(\xi_k)}\f$ instead, where the \f$\xi_k\f$ are
      33              :  * the collocation points. These are the coefficients this function returns.
      34              :  * Only for a unit weight function \f$w(x)=1\f$, i.e. a Legendre basis, is
      35              :  * \f$I[f]=Q[f]\f$ so this function returns the \f$w_k\f$ identically.
      36              :  *
      37              :  * For a `FiniteDifference` basis or `CellCentered` and `FaceCentered`
      38              :  * quadratures, the interpretation of the quadrature weights in term
      39              :  * of an approximation to \f$I(q)\f$ remains correct, but its explanation
      40              :  * in terms of orthonormal basis is not, i.e. we set \f$w_k\f$ to the grid
      41              :  * spacing at each point, and the inverse weight \f$\frac{1}{w(\xi_k)}=1\f$ to
      42              :  * recover the midpoint method for definite integrals.
      43              :  *
      44              :  * \param num_points The number of collocation points
      45              :  */
      46              : template <Basis BasisType, Quadrature QuadratureType>
      47            1 : const DataVector& quadrature_weights(size_t num_points);
      48              : 
      49              : /*!
      50              :  * \brief Quadrature weights for a one-dimensional mesh.
      51              :  *
      52              :  * \see quadrature_weights(size_t)
      53              :  */
      54            1 : const DataVector& quadrature_weights(const Mesh<1>& mesh);
      55              : }  // namespace Spectral