PolynomialInterpolation

class fridom.framework.grid.cartesian.polynomial_interpolation.PolynomialInterpolation(order: int = 1)

Bases: InterpolationModule

Polynomial interpolation for cartesian grids.

Description

Consider the following grid points:

\[x_i = (i - n/2) \Delta x, \quad i = 0, 1, \ldots, n\]

where \(n\) is the (odd) order of the polynomial interpolation. For example for \(n = 3\) we have the following grid points:

    We want to interpolate the field to this point (x=0)
                            ↓
        |   x_0   |   x_1   |   x_2   |   x_3   |
x/dx =     -3/2      -1/2       1/2       3/2

Let \(f_i\) be the field values at \(x_i\). We define the continuous extension of the field as:

\[f(x) = \sum_{i=0}^{n} \left( \prod_{j=0, j \neq i}^{n} \left( \frac{x - x_j}{x_i - x_j} f_i \right) \right)\]

By definition, \(f(x_i) = f_i\) holds. Finally, to interpolate the field to the point \(x=0\), we insert \(x=0\) into the above expression. Note that the grid spacing \(\Delta x\) cancels out.

\[f(0) = \sum_{i=0}^{n} c_i f_i\]

with the coefficients \(c_i\) given by:

\[c_i = \prod_{j=0, j \neq i}^{n} \frac{j-n/2}{j - i}\]

__init__(order: int = 1)

Methods

__init__([order])
disable()	Enabling the module means that it will be executed at each time step.
enable()	Enabling the module means that it will be executed at each time step.
interpolate(f, destination)	Interpolate the field to the destination position.
interpolate_axis(f, axis, destination)
is_enabled()	Return whether the module is enabled or not.
reset()	Stop and start the module.
setup(mset)	Start the module
start()	Start the module
stop()	Stop the module
update(mz)	Update the module

Attributes

diff_module	The differentiation module to be used by this module.
grid	The grid of the model settings
info	Return a dictionary with information about the time stepper.
interp_module	The interpolation module to be used by this module.
mset	The model settings
name
required_halo

name = 'Polynomial Interpolation'

setup(mset: ModelSettingsBase) → None

Start the module

Description

This method is called by the ModelSettings.setup() and sets the ModelSettings as well as the differentiation and interpolation modules.

interpolate(f: FieldVariable, destination: Position) → FieldVariable

Interpolate the field to the destination position.

Parameters

ffr.FieldVariable

The field to interpolate.

destinationfr.grid.Position

The position to interpolate to.

Returns

fr.FieldVariable

The interpolated field.

interpolate_axis(f: FieldVariable, axis: int, destination: AxisPosition) → FieldVariable