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 Quadrature : uint8_t;
      15              : enum class Parity : uint8_t;
      16              : }  // namespace Spectral
      17              : /// \endcond
      18              : 
      19              : namespace Spectral {
      20              : /// @{
      21              : /*!
      22              :  * \brief %Matrix used to compute the derivative of a function.
      23              :  *
      24              :  * \details For a function represented by the nodal coefficients \f$u_j\f$ a
      25              :  * matrix multiplication with the differentiation matrix \f$D_{ij}\f$ gives the
      26              :  * coefficients of the function's derivative. Since \f$u(x)\f$ is expanded in
      27              :  * Lagrange polynomials \f$u(x)=\sum_j u_j l_j(x)\f$ the differentiation matrix
      28              :  * is computed as \f$D_{ij}=l_j^\prime(\xi_i)\f$ where the \f$\xi_i\f$ are the
      29              :  * collocation points.
      30              :  *
      31              :  * The finite difference matrix uses summation by parts operators,
      32              :  * \f$D_{2-1}, D_{4-2}, D_{4-3}\f$, and \f$D_{6-5}\f$ from \cite Diener2005tn.
      33              :  *
      34              :  * \param num_points The number of collocation points
      35              :  */
      36              : template <Basis BasisType, Quadrature QuadratureType>
      37            1 : const Matrix& differentiation_matrix(size_t num_points);
      38              : template <Basis BasisType, Quadrature QuadratureType>
      39            1 : const Matrix& differentiation_matrix_transpose(size_t num_points);
      40              : /// @}
      41              : 
      42              : /*!
      43              :  * \brief %Matrix used to compute the derivative of a function of known parity
      44              :  *
      45              :  * \details For the Zernike basis, defined on \f$[0,1]\f$, `GaussRadauUpper`
      46              :  * quadratrue shifts the collocation points to the upper side, which
      47              :  * contributes to inaccurate differentiation at the lower side due to the low
      48              :  * density of points. By knowing the parity of functions in this basis as it
      49              :  * has two indices, we can extend the function to the negative \f$r\f$,
      50              :  * greatly reducing errors.
      51              :  *
      52              :  * \param num_points The number of collocation points
      53              :  * \param parity The Parity of the function
      54              :  */
      55              : template <Basis BasisType, Quadrature QuadratureType>
      56            1 : const Matrix& differentiation_matrix(size_t num_points, Parity parity);
      57              : 
      58              : /// @{
      59              : /*!
      60              :  * \brief Differentiation matrix for a one-dimensional mesh.
      61              :  *
      62              :  * \see differentiation_matrix(size_t)
      63              :  */
      64            1 : const Matrix& differentiation_matrix(const Mesh<1>& mesh);
      65            1 : const Matrix& differentiation_matrix_transpose(const Mesh<1>& mesh);
      66              : /// @}
      67              : 
      68              : /*!
      69              :  * \brief %Matrix used to compute the divergence of the flux in weak form.
      70              :  *
      71              :  * This is the transpose of the differentiation matrix multiplied by quadrature
      72              :  * weights that appear in DG integrals:
      73              :  *
      74              :  * \begin{equation}
      75              :  * \frac{D^T_{ij}} \frac{w_j}{w_i}
      76              :  * \end{equation}
      77              :  *
      78              :  * \param num_points The number of collocation points
      79              :  */
      80              : template <Basis BasisType, Quadrature QuadratureType>
      81            1 : const Matrix& weak_flux_differentiation_matrix(size_t num_points);
      82              : 
      83              : /*!
      84              :  * \brief %Matrix used to compute the divergence of the flux in weak form.
      85              :  *
      86              :  * \see weak_flux_differentiation_matrix(size_t)
      87              :  */
      88            1 : const Matrix& weak_flux_differentiation_matrix(const Mesh<1>& mesh);
      89              : }  // namespace Spectral