`jaxdf.operators.differential`

Operators

`derivative`

Given a field $u$, it returns the field

$$ \frac{\partial}{\partial \epsilon} u, \qquad \epsilon \in {x, y, \dots } $$

Concrete implementations:

`derivative(x: 'Continuous', *, axis = 0, params = None)`

Derivative operator for continuous fields.

Parameters:

Name	Type	Description	Default
`x`		Continuous field	required
`axis`		Axis along which to take the derivative	`0`

Returns:

Type	Description
	Continuous field

`diag_jacobian`

Given a vector field $u = (u_x,u_y,\dots)$ with the same dimensions as the dimensions of the domain, it returns the diagonal of the Jacobian matrix

$$ \left( \frac{\partial u_x}{\partial x}, \frac{\partial u_y}{\partial y}, \dots \right) $$

Concrete implementations:

`diag_jacobian(x: 'Continuous', *, params = None)`

Diagonal Jacobian operator for continuous fields.

Parameters:

Name	Type	Description	Default
`x`		Continuous field	required

Returns:

Type	Description
	The diagonal Jacobian of the field

`diag_jacobian(x: 'FiniteDifferences', *, stagger = [0], params = None)`

Diagonal Jacobian operator for finite differences fields.

Parameters:

Name	Type	Description	Default
`x`		FiniteDifferences field	required
`stagger`		Stagger of the derivative	`[0]`

Returns:

Type	Description
	The diagonal Jacobian of the field

`diag_jacobian(x: 'FourierSeries', *, stagger = [0], correct_nyquist = True, params = None)`

Diagonal Jacobian operator for Fourier series fields.

Parameters:

Name	Type	Description	Default
`x`	`FourierSeries`	Input field	required
`stagger`	`list`	Staggering value for the returned fields. The fields are staggered in the direction of their derivative. Defaults to [0].	`[0]`
`correct_nyquist`	`bool`	If `True`, uses a correction of the derivative filter for the Nyquist frequency, which preserves Hermitian symmetric and null space. See those notes for more details. Defaults to True.	`True`

Returns:

Type	Description
	The vector field whose components are the diagonal entries of the Jacobian of the input field.

`gradient`

Given a field $u$, it returns the vector field

$$ \nabla u = \left(\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \dots\right) $$

Concrete implementations:

`gradient(x: 'Continuous', *, params = None)`

Gradient operator for continuous fields.

Parameters:

Name	Type	Description	Default
`x`		Continuous field	required

Returns:

Type	Description
	The gradient of the field

`gradient(x: 'FiniteDifferences', *, stagger = [0], params = None)`

Gradient operator for finite differences fields.

Parameters:

Name	Type	Description	Default
`x`		FiniteDifferences field	required
`stagger`		Stagger of the derivative	`[0]`

Returns:

Type	Description
	The gradient of the field

`gradient(x: 'FourierSeries', *, stagger = [0], correct_nyquist = True, params = None)`

Gradient operator for Fourier series fields.

Parameters:

Name	Type	Description	Default
`x`	`FourierSeries`	Input field	required
`stagger`	`list`	Staggering value for the returned fields. The fields are staggered in the direction of their derivative. Defaults to [0].	`[0]`
`correct_nyquist`	`bool`	If `True`, uses a correction of the derivative filter for the Nyquist frequency, which preserves Hermitian symmetric and null space. See those notes for more details. Defaults to True.	`True`

Returns:

Type	Description
`FourierSeries`	The gradient of the input field.

`heterog_laplacian`

Given a field $u$ and a cofficient field $c$, it returns the field

$$ \nabla_c^2 u = \nabla \cdot (c \nabla u) $$

Concrete implementations:

`heterog_laplacian(x: 'FourierSeries', c: 'FourierSeries', *, params = None)`

Computes the position-varying laplacian using algorithm 4 of [Johnson, 2011].

Parameters:

Name	Type	Description	Default
`x`	`FourierSeries`	Input field	required
`c`	`FourierSeries`	Coefficient field	required

Returns:

Type	Description
	The Laplacian of the field

`laplacian`

Given a scalar field $u$, it returns the scalar field

$$ \nabla^2 u = \nabla \cdot \nabla u = \sum_{\epsilon \in {x,y,\dots}} \frac{\partial^2 u}{\partial \epsilon^2} $$

Concrete implementations:

`laplacian(x: 'Continuous', *, params = None)`

Laplacian operator for continuous fields.

Parameters:

Name	Type	Description	Default
`x`		Continuous field	required

Returns:

Type	Description
	The Laplacian of the field

`laplacian(x: 'FiniteDifferences', *, params = None)`

Gradient operator for finite differences fields.

Parameters:

Name	Type	Description	Default
`x`		FiniteDifferences field	required

Returns:

Type	Description
	The gradient of the field

`laplacian(x: 'FourierSeries', *, params = None)`

Laplacian operator for Fourier series fields.

Parameters:

Name	Type	Description	Default
`x`	`FourierSeries`	Input field	required

Returns:

Type	Description
	The Laplacian of the field

Utilities

`get_fd_coefficients(x, order=1, stagger=0)`

Returnst the stencil coefficients for a 1D Finite Differences derivative operator.

Parameters:

Name	Type	Description	Default
`x`	`FiniteDifferences`	FiniteDifferences field	required
`order`	`int`	Order of the derivative	`1`
`stagger`	`Union[float, int]`	Stagger of the derivative	`0`

Returns:

Type	Description
	Stencil coefficients

`fd_derivative_init(x, axis=0, stagger=0, *args, **kwargs)`

Initializes the stencils for FiniteDifferences derivatives. Accepts an arbitrary number of positional and keyword arguments after the mandatory arguments, which are ignored.

Parameters:

Name	Type	Description	Default
`x`	`FiniteDifferences`	FiniteDifferences field	required
`axis`		Axis along which to take the derivative	`0`
`stagger`		Stagger of the derivative	`0`

Returns:

Type	Description
	Stencil coefficients

`fd_diag_jacobian_init(x, stagger, *args, **kwargs)`

Initializes the parameters for the diagonal Jacobian of a FiniteDifferences field. Accepts an arbitrary number of positional and keyword arguments after the mandatory arguments, which are ignored.

Parameters:

Name	Type	Description	Default
`x`	`FiniteDifferences`	FiniteDifferences field	required
`stagger`		Stagger of the derivative	required

Returns:

Type	Description
	Stencil coefficients