Upwind_gradient.hpp File Reference

Approximating upwind gradients on a grid with the following options for the order of accuracy: 1, 3 or 5. More...

#include <cmath>
#include "Grid/grid_dist_id.hpp"
#include "FD_simple.hpp"
#include "Eno_Weno.hpp"

Go to the source code of this file.

Functions
template<typename field_type >
static field_type	upwinding (field_type dplus, field_type dminus, int sign)
	Upwinding: For a specific dimension, from the forward and backward gradient find the upwind side. More...
template<size_t Field, size_t Sign, typename gridtype , typename keytype >
auto	FD_upwind (gridtype &grid, keytype &key, size_t d, size_t order)
	Get the upwind finite difference of a scalar property on the current grid node. More...
template<size_t Field, size_t Sign, size_t Gradient, typename gridtype >
void	upwind_gradient (gridtype &grid, const bool one_sided_BC, size_t order)
	Computes upwind gradient with order of accuracy 1, 3 or 5. More...
template<typename gridtype >
static bool	ghost_width_is_sufficient (gridtype &grid, size_t required_width)
	Checks if ghost layer is thick enough for a given stencil-width. More...
template<size_t Field_in, size_t Sign, size_t Gradient_out, typename gridtype >
void	get_upwind_gradient (gridtype &grid, const size_t order=5, const bool one_sided_BC=true)
	Calls upwind_gradient. Computes upwind gradient of desired order {1, 3, 5} for the whole n-dim grid. More...

Detailed Description

Approximating upwind gradients on a grid with the following options for the order of accuracy: 1, 3 or 5.

For solving hyperbolic PDEs using differencing that is biased by the direction of the wave front. Upwinding is done according to M. Sussman and E. Fatemi, “Efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow” (1999) ), paragraph 4.1, 2 b). Order 1 is achieved by upwinding of forward and backward finite difference, order 3 and 5 by upwinding the ENO and WENO scheme, respectively.

Author

Justina Stark

Date

May 2021

Definition in file Upwind_gradient.hpp.

Function Documentation

FD_upwind()

template<size_t Field, size_t Sign, typename gridtype , typename keytype >

auto FD_upwind	(	gridtype &	grid,
		keytype &	key,
		size_t	d,
		size_t	order
	)

Get the upwind finite difference of a scalar property on the current grid node.

Order of accuracy can be 1, 3 or 5.

Template Parameters

Field	Size_t index of property for which the gradient should be computed.
Sign	Size_t index of property that contains the initial sign of that property for which the upwind FD should be computed.
gridtype	Type of input grid.
keytype	Type of key variable.

Parameters

grid	Grid, on which the gradient should be computed.
key	Key that contains the index of the current grid node.
d	Variable (size_t) that contains the dimension.
order	Order of accuracy of the difference scheme. Can be 1, 3 or 5.

Returns

Upwind finite difference in one dimension of the property under index Field on the current node with index key.

Definition at line 70 of file Upwind_gradient.hpp.

get_upwind_gradient()

template<size_t Field_in, size_t Sign, size_t Gradient_out, typename gridtype >

void get_upwind_gradient	(	gridtype &	grid,
		const size_t	order = 5,
		const bool	one_sided_BC = true
	)

Calls upwind_gradient. Computes upwind gradient of desired order {1, 3, 5} for the whole n-dim grid.

Template Parameters

Field	Size_t index of property for which the gradient should be computed.
Sign	Size_t index of property that contains the initial sign of that property for which the upwind FD should be computed / sign of the velocity.
Gradient	Size_t index of property where the upwind gradient result should be stored.
gridtype	Type of input grid.

Parameters

grid	Grid, on which the gradient should be computed.

Definition at line 217 of file Upwind_gradient.hpp.

ghost_width_is_sufficient()

template<typename gridtype >

static bool ghost_width_is_sufficient ( gridtype &  grid, size_t  required_width  )

static

Checks if ghost layer is thick enough for a given stencil-width.

Template Parameters

gridtype

Type of input grid.

Parameters

grid	Grid, for which ghost layer width should be checked.
required_width	Size_t variable, number of ghost nodes that are required in one direction.

Returns

Bool, true if ghost layer sufficiently large, false otherwise.

Definition at line 188 of file Upwind_gradient.hpp.

upwind_gradient()

template<size_t Field, size_t Sign, size_t Gradient, typename gridtype >

void upwind_gradient	(	gridtype &	grid,
		const bool	one_sided_BC,
		size_t	order
	)

Computes upwind gradient with order of accuracy 1, 3 or 5.

Checks if a point lays within the grid or at the boundary. For the internal grid points, it calls upwind finite difference FD_upwind(). For the border points, simply #FD_forward() and #FD_backward() is used, respectively, depending on the side of the border.

Template Parameters

Field	Size_t index of property for which the gradient should be computed.
Sign	Size_t index of property that contains the initial sign of that property for which the upwind FD should be computed.
Gradient	Size_t index of property where the gradient result should be stored.
gridtype	Type of input grid.

Parameters

grid	Grid, on which the gradient should be computed.
one_sided_BC	Bool variable, if true, use one-sided kernel for boundary-nodes. If false, extend stencil onto ghost nodes.
order	Size_t variable, order of accuracy the upwind FD scheme should have. Can be 1, 3 or 5.

Definition at line 123 of file Upwind_gradient.hpp.

upwinding()

template<typename field_type >

static field_type upwinding ( field_type  dplus, field_type  dminus, int  sign  )

static

Upwinding: For a specific dimension, from the forward and backward gradient find the upwind side.

Parameters

dplus	Gradient approximated using RHS neighbors.
dminus	Gradient approximated using LHS neighbors.
sign	Sign of the velocity with which the wave front is moving.

Returns

Scalar upwind gradient approximation in the dimension given.

Definition at line 39 of file Upwind_gradient.hpp.