Diffusion equation in TCLB solver¶

In this workshop we will cover basic conceptions related to TCLB solver. We will not dive into detail of TCLB compilation or model development - it is out of scope of this tutorial and is covered in other materials.

The d2q9_reaction_diffusion_system_XXXX models¶

This TCLB model is a collection of Reaction-Diffusion Partial/Ordinary differential equations systems. Basically, the model is build to solve systems in form:

\[\begin{split} \frac{\partial{\mathbf{DRE}_1(x,t)}}{\partial t} = \mathrm{Diffusivity\_DRE}_1 \nabla \mathbf{DRE}_1(x,t) + R_1(\mathbb{RHS}) \\ ... \\ \frac{\partial{\mathbf{DRE}_i(x,t)}}{\partial t} = \mathrm{Diffusivity\_DRE}_i \nabla \mathbf{DRE}_i(x,t) + R_i(\mathbb{RHS})\\ \frac{d{\mathbf{ODE}_1(x,t)}}{dt} = F_1(\mathbb{RHS}) \\ ... \\ \frac{d{\mathbf{ODE}_i(x,t)}}{dt} = F_i(\mathbb{RHS}) \end{split}\]

Where $DRE_1(x,t)$ is spatially variable field, solution of first Diffusion-Reaction equation, and $ODE_1(x,t)$ is solution of Ordinary Differential Equation at point $x$. Both equations are solved using second order, implicit midpoint scheme. Spatial discretisation of DRE is performed using LBM method - details could be found in [X]. The right hend side vector is equall $$ \mathbb{RHS} = \left[ DRE_i(x,t), ODE_i(x,t) \right] $$

Most of the models have meaningfull names for $\mathbf{DRE_i}$ variables and params, provided below.

At point of writing, supported model are:

d2q9_reaction_diffusion_system_SimpleDiffusion¶

\[ \frac{\partial{\mathbf{PHI}(x,t)}}{\partial t} = \mathbf{Diffusivity\_PHI} \nabla \mathbf{PHI}(x,t) \]

or

\[ \frac{\partial{\phi(x,t)}}{\partial t} = \nu \nabla \phi(x,t) \]

d2q9_reaction_diffusion_system_AllenCahn¶

\[ \frac{\partial{\mathbf{PHI}(x,t)}}{\partial t} = \mathbf{Diffusivity\_PHI} \nabla \mathbf{PHI}(x,t) + \mathbf{Lambda}*\mathbf{PHI}(x,t)*(1-\mathbf{PHI}(x,t)) \]

or

\[ \frac{\partial{\phi(x,t)}}{\partial t} = \nu \nabla \phi(x,t) + C\phi(1-\phi) \]

d2q9_reaction_diffusion_system_SIR_SimpleLaplace¶

This simple spatial extension of SIR model, by addition of diffusivity $$\nabla \left( S,I,R \right)$$ to every term

\[\begin{split} \frac{\partial}{\partial t} \mathbf{S} = - \mathbf{Beta} * \mathbf{S}* \mathbf{I} + \mathbf{Diffusivity\_S} \nabla \mathbf{S}\\ \frac{\partial}{\partial t} \mathbf{I} = \mathbf{Beta} * \mathbf{S}* \mathbf{I} - \mathbf{Gamma} * \mathbf{I} + \mathbf{Diffusivity\_I} \nabla \mathbf{I} \\ \frac{\partial}{\partial t} \mathbf{R} = \mathbf{Gamma}* \mathbf{I} + \mathbf{Diffusivity\_R} \nabla \mathbf{R} \end{split}\]

or

\[\begin{split} \frac{\partial}{\partial t} S = - \beta \frac{S}{N}I + \nu_S \nabla S\\ \frac{\partial}{\partial t} I = \beta \frac{S}{N}I - \gamma I + \nu_I \nabla I \\ \frac{\partial}{\partial t} R = \gamma I + \nu_R \nabla R \end{split}\]

for $N=1$

d2q9_reaction_diffusion_system_SIR_ModifiedPeng¶

or the WSIR model, aleready referenced in this workshop

\[\begin{split} \frac{\partial}{\partial t} \mathbf{W} = \mathbf{Beta_w} * \left[(\mathbf{I}-\mathbf{W}) \right] + \mathbf{Diffusivity\_W} \nabla \mathbf{W}\\ \frac{\partial}{\partial t} \mathbf{S} = - \mathbf{Beta} * \frac{\mathbf{S}*\mathbf{W}}{\mathbf{N}}* \\ \frac{\partial}{\partial t} \mathbf{I} = \mathbf{Beta} * \frac{\mathbf{S}*\mathbf{W}}{\mathbf{N}} - \mathbf{Gamma} * \mathbf{I} \\ \frac{\partial}{\partial t} \mathbf{R} = \mathbf{Gamma}* \mathbf{I} \end{split}\]

or

\[\begin{split} \frac{\partial}{\partial t} W = \beta_W \left[\frac{r^2}{8}W + (I-W) \right] \\ \frac{\partial}{\partial t} S = - \beta \frac{S}{N}W\\ \frac{\partial}{\partial t} I = \beta \frac{S}{N}W - \gamma I \\ \frac{\partial}{\partial t} R = \gamma I \end{split}\]

Keep in mind that for consistency, it’s recomended to set $$ \mathbf{Diffusivity\_W} = \mathbf{Beta_w} * \frac{r^2}{8} $$

Additionally, list of models could be extracted via:

! tclbmake d2q9_reaction_diffusion_system/list

TCLB_PATH:  /home/mdzik/_TCLB/TCLB
                                                 AllenCahn SIR_ModifiedPeng
d2q9_reaction_diffusion_system_AllenCahn                 X                -
d2q9_reaction_diffusion_system_SIR_ModifiedPeng          -                X
d2q9_reaction_diffusion_system_SIR_SimpleLaplace         -                -
d2q9_reaction_diffusion_system_SimpleDiffusion           -                -
                                                 SIR_SimpleLaplace
d2q9_reaction_diffusion_system_AllenCahn                         -
d2q9_reaction_diffusion_system_SIR_ModifiedPeng                  -
d2q9_reaction_diffusion_system_SIR_SimpleLaplace                 X
d2q9_reaction_diffusion_system_SimpleDiffusion                   -
                                                 SimpleDiffusion
d2q9_reaction_diffusion_system_AllenCahn                       -
d2q9_reaction_diffusion_system_SIR_ModifiedPeng                -
d2q9_reaction_diffusion_system_SIR_SimpleLaplace               -
d2q9_reaction_diffusion_system_SimpleDiffusion                 X

What is included in this environment¶

To run TCLB model us ipython shell binding:

! tclb ModelName ConfigurationFile.xml

This will effectivelu run (in the curren working directory)

$TCLB_PATH/CLB/ModelName/main ConfigurationFile.xml

in your Jupyter/Binder. We will use

! tclb ModelName ConfigurationFile.xml  > /dev/null && echo "DONE!"

to supress output in most cases.

TCLB solver¶

As you might noticed TCLB is a set of models (like d2q9). TCLB uses XML as source of configuration plus limited numebr of command line options which will be covered latter. There is a python library that helps you build XML on-the-fly if You like.

import h5py
import numpy as np
import matplotlib.pyplot as plt
from display_xml import XML

Modelling of epidemic spreading with Lattice Boltzmann Method

Diffusion equation in TCLB solver

Contents

Diffusion equation in TCLB solver¶

The d2q9_reaction_diffusion_system_XXXX models¶

d2q9_reaction_diffusion_system_SimpleDiffusion¶

d2q9_reaction_diffusion_system_AllenCahn¶

d2q9_reaction_diffusion_system_SIR_SimpleLaplace¶

d2q9_reaction_diffusion_system_SIR_ModifiedPeng¶

What is included in this environment¶

TCLB solver¶

Basic configuration file¶

and let’s run it:¶

Using TCLBConfig writer class¶

RInside: periodic IC and comparison with Finite Difference¶

Let’s create initial “data file”¶

Now the XML configuration¶

Solution obtained from TCLB¶

Comparison with Backward Time - Central Space FD¶

RInside: read an image as initial condition¶

Let’s check if it work - we use an image of a M-18A Dromader airplane as initial condition ;)¶

The CLBc object could be reused (something could be appended to it after save, parameters could bo overwritten)¶