2.7. drrc.spatially_extended_systems module

Solver for spatially extended systems.

Code author: Luk Fleddermann

class LocalModel(Pardict)

Bases: object

Local Model of Exitation.

The init function creates the environment for the different models of the electric excitation, i.e. it sets parameters for differential equation and a propper Grid-size (Physical and numerical) for the animation.

Parameters:

Pardict – Parameterfile

fitzhugh_nagumo_local(vars)

The local FitzHugh-Nagumo model follows

\[\begin{split}\partial_t u(x,y,t) &= au(u-b)(1-u)-w\,,\\ \partial_t w(x,y,t) &= -\varepsilon(u-w)\,,\end{split}\]

which is extended to the spatially extended FitzHugh-Nagumo model by adding a diffusion term to the u variable.

Parameters:

vars (np.ndarray) – excitation of variables [u,w] = [vars[0], vars[1]]

Returns:

change in excitation of u and w following the fitz_nagumo model. If LocalModel is of wrong type, returns -1.

Return type:

np.ndarray

Notes

The class needs to be initialized, with model='fitzhugh_nagumo, for the function to work.

aliev_panfilov_local(vars)

The local Aliev-Panfilov model follows

\[\begin{split}\partial_t u(x,y,t) &= ku(u-a)(1-u)-uw\,,\\ \partial_t w(x,y,t) &= \left(\varepsilon+\frac{\mu_1 w}{\mu_2 +u}\right)\cdot(-w-ku(u-b-1))\,,\end{split}\]

which is extended to the spatially extended Aliev-Panfilov model by adding a diffusion term to the u variable.

Parameters:

vars (np.ndarray) – excitation of variables

Returns:

change in excitation of u and w following the fitz_nagumo model. If class.model is of wrong type, returns -1.

Return type:

np.ndarray

Notes

The class needs to be initialized, with model='aliev_panfilov, for the function to work.

class Diffusion(Pardict)

Bases: object

Diffusion model, different accuracy (5 point/ 9 point stencil). May be extended to other methods of implementing diffusion.

Parameters:

• model (str) – Specification of the model of partial differential eq. one wants to use

• stencil (str) – Specification of the stencil used in the calculation of the Laplacian

• stable (bool) – Choice whether stable or chaotic spiral waves are produced with the Aliev-Panfilov model

apply_5stencil(u)

Parameters:

u – excitation of grid.

Returns:

Two dimensional array of discrete Laplacian of the excitation of u, using the five-point stencil.

Return type:

laplacian

Notes

The outer layer of u needs to be a ghost layer s.t. no flux boundary conditions are inforced. Without ghost layer periodic boundary conditions are used.

apply_9stencil(u)

Parameters:

u – excitation of grid.

Returns:

Two dimensional array of discrete Laplacian of the excitation of u, using the five-point stencil.

Return type:

laplacian

Notes

The outer layer of u needs to be a ghost layer s.t. no flux boundary conditions are inforced. Without ghost layer periodic boundary conditions are used.

class BoundaryCondition(Pardict)

Bases: object

Different Typs of Boundary conditions. So far only ‘no_flux’ has a proper meaning.

The init function creates the environment for the different models of the electric excitation, i.e. it sets parameters for differential equation and a proper Grid-size (Physical and numerical) for the animation.

Parameters:

• model (str) – Specification of the model of partial differential eq. one wants to use

• stencil (str) – Specification of the stencil used in the calculation of the Laplacian

• stable (bool) – Choice wether stable or chaotic spiral waves are produced with the Aliev-Panfilov model

no_flux(vars, itterator=None)

Parameters:

u – excitation of grid.

Returns:

excitation of grid with no flux boundary condition enforced for 5-point stencil.

Return type:

u

Notes

The outer layer is turned in a ghost layer with no physical meaning. The corners of the grid (i.e. u[0,0]) are unimportant for the calculations of the five point stencil, hence orientation of the four executed steps does not matter.

class IntegrationMethods

Bases: object

Methods of temporal integraion of class ‘LocalModel’ coupled by class ‘Diffusion’.

static exp_euler_itt(dif: Diffusion, lm: LocalModel, bc: BoundaryCondition, Pardict: dict, seed: int | None = None)

Iterative application of explicit Euler method.

Parameters:

• dif – Diffusion Model

• lm – Local Model

• bc – Boundary Conditions

• Pardict – Parameter File

• seed – seed of random initial condition

Returns:

Last time step of integration

Return type:

vars

static exp_euler_itt_with_save(dif: Diffusion, lm: LocalModel, bc: BoundaryCondition, Pardict: dict, vars0: ndarray | None = None, seed: int | None = None, prog_feetback: bool = True)

Iterative application of explicit euler method.

Parameters:

• dif – Diffusion Model

• lm – Local Model

• bc – Boundary Conditions

• Pardict – Parameter File

• vars0 – Initial Condition

• seed – seed of random initial condition

Returns:

Last time step of integration

Return type:

vars

static _get_initial(Pardict: dict, vars0: ndarray | None = None, seed: int | None = None) → ndarray

Creates initial condition (array) from string in dictionary.

Parameters:

• Pardict (dict) – initial condition

• vars0 (np.ndarray) – initial condition

• seed (int) – seed of random initial condition

Returns:

initial condition

Return type:

np.ndarray

Notes

Pardict needs to have keys ‘LocalModel’, ‘SpatialParameter’ and ‘initial_condition’