## 3 Numerical Formulations for the Normalized Fokker-Planck Equation
### Discretization of the Model Equation  
The heat conduction equation is a parabolic PDE with the form  
\[\frac{\partial u}{\partial t}=\rho\frac{\partial^{2}u}{\partial x^{2}} \tag{23}\]  
which is typically characterized by a diffusivity (or dispersion) coefficient \(\rho\). The convective (or advection) equation such as the wave equation is categorized as a hyperbolic type given by  
\[\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial x}=0 \tag{24}\]  
Consider the one-dimensional transport equation with both convection and diffusion terms,  
\[\frac{\partial u}{\partial t}=-v\frac{\partial u}{\partial x}+\rho\frac{ \partial^{2}u}{\partial x^{2}},\text{ where }v=\text{const},\;\rho=\text{const}>0 \tag{25}\]  
where \(u\) is a scalar, being convected with a known velocity \(v\) and being diffused. Equation (25) is a one-dimensional non-linear convective diffusion equation, or a hyperbolic parabolic equation; it is hyperbolic due to the convective term, \(v\partial u/\partial x\), and it is parabolic due to the diffusive term, \(\rho\partial^{2}u/\partial x^{2}\). Like the diffusion equation the transport equation is strictly parabolic. Closed-form solution \(u(x,t)\) of Equation (25) is usually not available except for very simplified cases. It can, however, be obtainednumerically in most applications, for example, by the finite difference and the finite element methods. The transport equation, as the model equation in this paper, will be implemented. Explicit methods are easier to solve than implicit methods in terms of time and effort spent on a computer. However, there exist several disadvantages in explicit methods. For the explicit finite difference method to be stable, the grid spacing in the time domain and grid spacing in the space domain need to satisfy certain requirements. The choice of grid spacing is made in accordance with the condition of stability. Implicit methods offer great advantage on the stability of the finite difference equations, since most are unconditionally stable. Therefore, a larger grid size in time is permitted. It should be noted that the selection of a larger time step is limited due to accuracy consideration because an increase in time step will increase the truncation error of the finite difference equation. In the finite-difference methods, partial derivatives are approximated using Taylor's series; hence, solutions can be obtained using the first-, second-, or higher-order accurate methods. In this paper, the following finite difference approximations are selected for the partial derivatives:  
\[\frac{\partial u(x,t)}{\partial t}=\frac{u(x,t+\Delta t)-u(x,t)}{\Delta t}+O( \Delta t)\approx\frac{u_{j}^{n+1}-u_{j}^{n}}{\Delta t} \tag{26}\]  
\[\frac{\partial u(x,t)}{\partial x}=\frac{u(x+\Delta x,t)-u(x-\Delta x,t)}{2 \Delta x}+O(\Delta x)^{2}\approx\frac{u_{j+1}^{n+1}-u_{j-1}^{n+1}}{2\Delta x} \tag{27}\]  
\[\frac{\partial^{2}u(x,t)}{\partial x^{2}}=\frac{u(x+\Delta x,t)-2u(x,t)+u(x- \Delta x,t)}{\left(\Delta x\right)^{2}}+O(\Delta x)^{2}\approx\frac{u_{j+1}^{ n+1}-2u_{j}^{n+1}+u_{j-1}^{n+1}}{\left(\Delta x\right)^{2}} \tag{28}\]  
An equation is defined as being implicit when more than one unknown appears in the finite difference equation. For the implicit methods, a set of simultaneous equations needs to be solved, which requires more computation time per time step for the implicit methods. Implicit methods generally possess better stability properties and higher accuracy than the explicit methods. For implicit formulation, most are unconditionally stable. Implicit methods include (1) the backward time/central space (BTCS) implicit method, and (2) the Crank-Nicolson (CN) method.