2.3 Introduction to Finite Difference Methods

2.3.2 Finite Difference Methods

To approximate the convection-diffusion equation we can combine various finite difference derivative approximations. For example, consider the one-dimensional convection-diffusion equation,

  \begin{equation} \frac{\partial U}{\partial t} + u\frac{\partial U}{\partial x} - \mu\frac{\partial^2 U}{\partial x^2} = 0 \label{equ:condif1d_pde} \end{equation}   (2.55)

Approximating this using central differences for all derivatives, the convection-diffusion equation at node \( i \) can be approximated as,

  \begin{equation} \frac{{\rm d}U_i}{{\rm d}t} + u_{i}\delta_{2x} U_{i} - \mu \delta^2_x U_{i} = 0 \label{equ:condif_central} \end{equation}   (2.56)

This is an ordinary differential equation for \( U_{i} \) which is coupled to the nodal values at \( U_{i\pm 1} \). We refer to Equation 2.33 as being semi-discrete, since we have discretized the PDE in space but not in time. To make this a fully discrete approximation, we need to discretize in time. To do this, we could apply any of the ODE integration methods that we discussed previously. For example, the simple forward Euler integration method would give,

  \begin{equation} \frac{U_i^{n+1}-U_i^n}{\Delta t} + u_{i}^n\delta_{2x} U^n_{i} = \mu \delta^2_x U^n_{i}. \label{equ:condif_ftcs} \end{equation}   (2.57)

Using central difference operators for the spatial derivatives and forward Euler integration gives the method widely known as a Forward Time-Central Space (FTCS) approximation.