README.md

poisson_lbm

A small collection of 1D Poisson solvers based on the lattice Boltzmann method

Description

This package contains an implementation of the one-dimensional Poisson solver described in the paper: Chai, Z., & Shi, B. (2008). A novel lattice Boltzmann model for the Poisson equation. Applied mathematical modelling, 32(10), 2050-2058.

Examples

Usage examples can be found in the apps folder.

In the first example we solve the Poisson-Boltzmann equation with Debye-Huckel approximation as described in the original paper. The equation is given as

with the boundary conditions

The analytical solution of this problem is given by

A plot of the analytical and numerical solutions is shown below:

In the second example we solve a steady-state reaction diffusion problem. For a first-order reaction in a catalyst slab we can derive the following equation:

and boundary conditions

The value Th is known as the Thiele modulus and represents the ratio between the reaction rate and the diffusion rate. An analytical solution for this problem is given by

The symmetry (or zero-flux) boundary condition can be implemented with a second-order one-sided finite difference. A plot of the agreement between analytical and numerical solutions is shown below: