AdamBashforth

class fridom.framework.time_steppers.AdamBashforth(dt=1, order: int = 3, eps=0.01)

Bases: TimeStepper

Adam Bashforth time stepping up to 4th order.

Parameters

dtfloat

Time step size. (default 0.01)

orderint

Order of the time stepping. (default 3, max 4)

epsfloat

2nd order bashforth correction. (default 0.01)

Description

The Adam Bashforth time stepping scheme is a multi-step explicit time stepping scheme. It solves a given PDE

\[\partial_t \boldsymbol{z} = \boldsymbol{F}(\boldsymbol{z}, t)\]

by using the following scheme of order \(n\)

\[\boldsymbol{z}^{n+1} = \boldsymbol{z}^n + \Delta t \sum_{j=0}^{n-1} \alpha_j \boldsymbol{F}(\boldsymbol{z}^{n-j}, t^{n-j})\]

where \(\alpha_i\) are the Adam Bashforth coefficients, \(\Delta t\) is the time step size, \(\boldsymbol{z}^j\) is the state at time \(t^j = t_0 + j \Delta t\). The coefficients for orders 1 to 4 are given in the table below.

Order	\(\alpha_1\)	\(\alpha_2\)	\(\alpha_3\)	\(\alpha_4\)
1	1
2	3/2 + epsilon	-1/2 - epsilon
3	23/12	-4/3	5/12
4	55/24	-59/24	37/24	-3/8

Stability Analysis

Let \(\lambda\) be the eigenvalues of the right-hand side of the PDE, e.g:

\[\partial_t \boldsymbol{z} = \boldsymbol{F}(\boldsymbol{z}, t) = -i \lambda \boldsymbol{z}\]

Inserting this into the Adam Bashforth scheme gives:

\[\boldsymbol{z}^{n+1} = \sum_{j=0}^{n-1} c_j \boldsymbol{z}^{n-j}\]

where

\[\begin{split}c_j = \begin{cases} 1 - i \Delta t \lambda & \text{if } j = 0 \\ -i \Delta t \lambda & \text{if } j > 0 \end{cases}\end{split}\]

We now insert the Ansatz:

\[\boldsymbol{z}^n = \boldsymbol{z}_0 e^{-i \omega n \Delta t} = \boldsymbol{z}_0 x^n\]

with \(x = e^{-i \omega \Delta t}\). This yields a polynomial equation for \(x\):

\[x^{n+1} = \sum_{j=0}^{n-1} c_j x^{n-j}\]

Finally, we find the eigenvalues of the time stepping scheme by solving the polynomial equation for \(x\) numerically and taking the logarithm:

\[\omega = -i \log(x) / \Delta t\]

__init__(dt=1, order: int = 3, eps=0.01)

Methods

__init__([dt, order, eps])
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.
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
time_discretization_effect(omega)	Compute the time discretization effect on a frequency.
update(mz)	Update the time stepper.
update_coeff_AB()	Upward ramping of Adam-Bashforth coefficients after restart.
update_pointer()	Update pointer for Adam-Bashforth time stepping.
update_tendency()

Attributes

diff_module	The differentiation module to be used by this module.
dt	Time step size.
dz	Returns a pointer on the current tendency term.
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 = 'Adam Bashforth'

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.

reset()

Stop and start the module.

update(mz: ModelState)

Update the time stepper.

update_tendency()

update_pointer() → None

Update pointer for Adam-Bashforth time stepping.

update_coeff_AB() → None

Upward ramping of Adam-Bashforth coefficients after restart.

time_discretization_effect(omega: ndarray) → ndarray

Compute the time discretization effect on a frequency.

Parameters

omegandarray | float | complex

The frequency of the wave.

Returns

ndarray

The frequency of the wave including the time discretization effect.

property dt: timedelta64

Time step size.

property info: dict

Return a dictionary with information about the time stepper.

Description

This method should be overridden by the child class to return a dictionary with information about the time stepper. This information is used to print the time stepper in the __repr__ method.

property dz

Returns a pointer on the current tendency term.