Ccz4::fd Namespace Reference

The finite difference second derivatives for implementing the CCZ4 system. More...

Functions
template<size_t Dim>
void	second_logical_partial_derivatives (gsl::not_null< std::array< gsl::span< double >, Dim > * > pure_second_logical_derivatives, gsl::not_null< std::array< gsl::span< double >, Dim > * > mixed_second_logical_derivatives, const gsl::span< const double > &volume_vars, const DirectionMap< Dim, gsl::span< const double > > &ghost_cell_vars, const Mesh< Dim > &volume_mesh, size_t number_of_variables, size_t fd_order)
	Compute the pure and mixed second logical partial derivatives using finite difference derivatives. Only 3 dimensions 4th-order fd is supported. More...
template<typename DerivativeTags , size_t Dim, typename DerivativeFrame >
void	second_partial_derivatives (gsl::not_null< Variables< db::wrap_tags_in< Tags::second_deriv, DerivativeTags, tmpl::size_t< Dim >, DerivativeFrame > > * > second_partial_derivatives, const gsl::span< const double > &volume_vars, const DirectionMap< Dim, gsl::span< const double > > &ghost_cell_vars, const Mesh< Dim > &volume_mesh, size_t number_of_variables, size_t fd_order, const InverseJacobian< DataVector, Dim, Frame::ElementLogical, DerivativeFrame > &inverse_jacobian)
	Compute the second partial derivatives on the DerivativeFrame using the inverse_jacobian and inverse_hessian. Only 3 dimensions 4th-order fd is supported. More...

Detailed Description

The finite difference second derivatives for implementing the CCZ4 system.

Function Documentation

second_logical_partial_derivatives()

template<size_t Dim>

void Ccz4::fd::second_logical_partial_derivatives	(	gsl::not_null< std::array< gsl::span< double >, Dim > * >	pure_second_logical_derivatives,
		gsl::not_null< std::array< gsl::span< double >, Dim > * >	mixed_second_logical_derivatives,
		const gsl::span< const double > &	volume_vars,
		const DirectionMap< Dim, gsl::span< const double > > &	ghost_cell_vars,
		const Mesh< Dim > &	volume_mesh,
		size_t	number_of_variables,
		size_t	fd_order
	)

Compute the pure and mixed second logical partial derivatives using finite difference derivatives. Only 3 dimensions 4th-order fd is supported.

We use a symmetric stencil in the bulk and slightly asymetric stencil for the mixed partial derivatives where ghost data from two neighbors is needed (to avoid communication across diagonal elements). The mixed_second_logical_derivatives return an 3D array corresponding to xy, yz, xz mixed partial derivatives.

The stencil for the pure second derivatives is

\[ \frac{\partial^2 f}{\partial x^2} = - \frac{1}{12\delta^2}[f(x+2\delta,y)+f(x-2\delta,y)] + \frac{4}{3\delta^2}[f(x+\delta,y)+f(x-\delta,y)]-\frac{5}{2\delta^2}f(x,y) + O(\delta^4). \]

The stencil for the mixed second derivatives in the bulk is

\[ \begin{aligned} \frac{\partial^2 f}{\partial x \partial y} =& \frac{1}{3\delta^2}[f(x+\delta,y+\delta) + f(x-\delta,y-\delta) - f(x+\delta,y-\delta) - f(x-\delta,y+\delta)] \\ &+ \frac{1}{48\delta^2}[f(x+2\delta,y-2\delta) + f(x-2\delta,y+2\delta) - f(x+2\delta,y+2\delta) - f(x-2\delta,y-2\delta)] + O(\delta^4). \end{aligned}\]

The stencil for the mixed second derivatives in the ghost zone is (with descending diagonal)

\[ \begin{aligned} \frac{\partial^2 f}{\partial x \partial y} =& \frac{1}{24\delta^2}[f(x+2\delta,y-2\delta) + f(x-2\delta,y+2\delta) - f(x+2\delta,y) - f(x-2\delta,y) - f(x,y+2\delta) - f(x,y-2\delta)] \\ &+ \frac{2}{3\delta^2}[f(x+\delta,y) + f(x-\delta,y) + f(x,y+\delta) + f(x,y-\delta) - f(x+\delta,y-\delta) - f(x-\delta,y+\delta)] - \frac{5}{4\delta^2}f(x,y) + O(\delta^4), \end{aligned}\]

and the ascending diagonal stencil has the opposite weights.

Note

Currently the stride is always one because we transpose the data before reconstruction.

second_partial_derivatives()

template<typename DerivativeTags , size_t Dim, typename DerivativeFrame >

void Ccz4::fd::second_partial_derivatives	(	gsl::not_null< Variables< db::wrap_tags_in< Tags::second_deriv, DerivativeTags, tmpl::size_t< Dim >, DerivativeFrame > > * >	second_partial_derivatives,
		const gsl::span< const double > &	volume_vars,
		const DirectionMap< Dim, gsl::span< const double > > &	ghost_cell_vars,
		const Mesh< Dim > &	volume_mesh,
		size_t	number_of_variables,
		size_t	fd_order,
		const InverseJacobian< DataVector, Dim, Frame::ElementLogical, DerivativeFrame > &	inverse_jacobian
	)

Compute the second partial derivatives on the DerivativeFrame using the inverse_jacobian and inverse_hessian. Only 3 dimensions 4th-order fd is supported.

First and second logical partial derivatives are first computed using the fd::logical_partial_derivatives() and Ccz4::fd::second_logical_partial_derivatives() function.

Note

The inverse_hessian has not been implemented, so currently inertial coordinates cannot mix logical coordinates.