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