A small collection of 1D Poisson solvers based on the lattice Boltzmann method
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.
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 $$\nabla^2 u = k^2 u$$ with the boundary conditions $$u(0)=1, \quad u(1)=1.$$
The analytical solution of this problem is given by $$u(x) = \left(\frac{e^{k} - 1}{e^{k} - e^{-k}}\right) e^{-kx} + \left(\frac{1 - e^{-k}}{e^{k} - e^{-k}}\right) e^{kx}.$$ 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:
$$\nabla^2 u = \mathit{Th}^2 u$$ and boundary conditions $$u(0)=1, \quad \frac{\partial u}{\partial x}|_{x=1}=0.$$
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 $$u(x) = \frac{\cosh\left(\mathit{Th}(1-x)\right)}{\cosh(\mathit{Th})}.$$
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:
