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 Matrix;
      10             : template <size_t>
      11             : class Mesh;
      12             : namespace Spectral {
      13             : enum class Basis : uint8_t;
      14             : enum class Parity : uint8_t;
      15             : enum class Quadrature : uint8_t;
      16             : }  // namespace Spectral
      17             : /// \endcond
      18             : 
      19             : namespace Spectral {
      20             : /*!
      21             :  * \brief %Matrix used to transform from the nodal coefficients of a function to
      22             :  * its spectral coefficients (modes). Also referred to as the inverse
      23             :  * _Vandermonde matrix_.
      24             :  *
      25             :  * \details This is the inverse to the Vandermonde matrix \f$\mathcal{V}\f$
      26             :  * computed in modal_to_nodal_matrix(size_t). It can be computed
      27             :  * analytically for Gauss quadrature by evaluating
      28             :  * \f$\sum_j\mathcal{V}^{-1}_{ij}u_j=\widetilde{u}_i=
      29             :  * \frac{(u,\Phi_i)}{\gamma_i}\f$
      30             :  * for a Lagrange basis function \f$u(x)=l_k(x)\f$ to find
      31             :  * \f$\mathcal{V}^{-1}_{ij}=\mathcal{V}_{ji}\frac{w_j}{\gamma_i}\f$ where the
      32             :  * \f$w_j\f$ are the Gauss quadrature weights and \f$\gamma_i\f$ is the norm
      33             :  * square of the spectral basis function \f$\Phi_i\f$.
      34             :  *
      35             :  * \param num_points The number of collocation points
      36             :  *
      37             :  * \see modal_to_nodal_matrix(size_t)
      38             :  */
      39             : template <Basis BasisType, Quadrature QuadratureType>
      40           1 : const Matrix& nodal_to_modal_matrix(size_t num_points);
      41             : 
      42             : /*!
      43             :  * \brief %Matrix used to transform from the nodal coefficients of a function to
      44             :  * its spectral coefficients (modes). This two-index version is used for
      45             :  * two-dimensional basis function (i.e. Zernike with GaussRadauUpper
      46             :  * quadrature).
      47             :  *
      48             :  * For Zernike, \f$m\f$ is the angular index and \f$N\f$ is the
      49             :  * maximum supported spectral index (usually taken to be the maximum value,
      50             :  * \f$2 \, \texttt{num_points}-2\f$). Note that the size of a spectral space
      51             :  * vector is \f$(N - m) / 2 + 1\f$, using integer division.
      52             :  *
      53             :  * \see nodal_to_modal_matrix(size_t)
      54             :  */
      55             : template <Basis BasisType, Quadrature QuadratureType>
      56           1 : const Matrix& nodal_to_modal_matrix(size_t num_points, size_t m, size_t N);
      57             : 
      58             : /*!
      59             :  * \brief Transformation matrix from nodal to modal coefficients for a
      60             :  * one-dimensional mesh.
      61             :  *
      62             :  * \see nodal_to_modal_matrix(size_t)
      63             :  */
      64           1 : const Matrix& nodal_to_modal_matrix(const Mesh<1>& mesh);
      65             : 
      66             : /*!
      67             :  * \brief Transformation matrix from nodal to modal coefficients for a
      68             :  * one-dimensional mesh with a Zernike basis.
      69             :  *
      70             :  * \see nodal_to_modal_matrix(size_t, size_t, size_t)
      71             :  */
      72           1 : const Matrix& nodal_to_modal_matrix(const Mesh<1>& mesh, size_t m, size_t N);
      73             : 
      74             : /*!
      75             :  * \brief %Matrix used to transform from the nodal coefficients of a function to
      76             :  * its spectral coefficients (modes), for a parity-dependent basis such as
      77             :  * `Basis::HalfFourier`.
      78             :  *
      79             :  * \param num_points The number of collocation points
      80             :  * \param parity The parity of the function (`Parity::Even` or `Parity::Odd`)
      81             :  *
      82             :  * \see nodal_to_modal_matrix(size_t)
      83             :  */
      84             : template <Basis BasisType, Quadrature QuadratureType>
      85           1 : const Matrix& nodal_to_modal_matrix(size_t num_points, Parity parity);
      86             : 
      87             : /*!
      88             :  * \brief Transformation matrix from nodal to modal coefficients for a
      89             :  * one-dimensional mesh with a parity-dependent basis such as
      90             :  * `Basis::HalfFourier`.
      91             :  *
      92             :  * \see nodal_to_modal_matrix(size_t, Parity)
      93             :  */
      94           1 : const Matrix& nodal_to_modal_matrix(const Mesh<1>& mesh, Parity parity);
      95             : }  // namespace Spectral