# 2.3 Introduction to Finite Difference Methods

## 2.3.1 Finite Difference Approximations

Recall how the multi-step methods we developed for ODEs are based on a truncated Taylor series approximation for $$\frac{\partial U}{\partial t}$$. Specifically, we can consider each multi-step method as computing $$U$$ at discrete instances in time $$(t^0, t^1, \ldots , t^ n, \ldots )$$ where the derivative $$\left. \frac{\partial U}{\partial t}\right|_ n = U_ t^ n$$ is approximated using a combination of $$\left(U^{n+1}, U^ n, U^{n-1}, \ldots \right)$$.

Finite difference methods for PDEs are essentially built on the same idea, but approximating spatial derivatives instead of time derivatives. Namely, the solution $$U$$ is approximated at discrete instances in space $$\left(x_0, x_1, \ldots , x_{i-1}, x_ i, x_{i+1}, \ldots , x_{N_ x-1}, x_{N_ x}\right)$$ where the spatial derivatives $$\left( \left.\frac{\partial U}{\partial x} \right|_ i = U_{x_ i}, \left. \frac{\partial ^2 U}{\partial x^2} \right|_ i = U_{xx_ i} \right)$$ are approximated using a combination of $$(U_ i, U_{i \pm 1}, U_{i \pm 2}, \ldots )$$.

Figure 2.8: One-dimensional structured mesh for finite difference approximations.

In the following we derive a finite difference approximation of $$\frac{\partial U}{\partial x}$$ at node $$x_ i$$. For a differentiable function $$U$$, the derivative at the point $$x_ i$$ is given by:

 $\left. \frac{\partial U}{\partial x} \right|_{x_ i} = \lim _{\Delta x \to 0} \frac{U(x_ i+\Delta x) - U(x_ i - \Delta x)}{2\Delta x}.$ (2.45)

The finite difference approximation is obtained by eliminating the limiting process:

 ${U_ x}_ i \approx \frac{U(x_ i+\Delta x) - U(x_ i-\Delta x)}{2 \Delta x} = \frac{U_{i+1} - U_{i-1}}{2 \Delta x} \equiv \delta _{2x} U_ i.$ (2.46)

The finite difference operator $$\delta _{2x}$$ is called a central difference operator. Finite difference approximations can also be one-sided. For example, a backward difference approximation is,

 $\left.\frac{\partial U}{\partial x}\right|_{i,j} \approx \delta _{x}^{-} U_{i,j} \equiv \frac{1}{{\scriptstyle \Delta } x}\left(U_{i,j} - U_{i-1,j}\right), \label{equ:ux_ backwarddiff}$ (2.47)

and a forward difference approximation is,

 $\left.\frac{\partial U}{\partial x}\right|_{i,j} \approx \delta _{x}^{+} U_{i,j} \equiv \frac{1}{{\scriptstyle \Delta } x}\left(U_{i+1,j} - U_{i,j}\right), \label{equ:ux_ forwarddiff}$ (2.48)

We can also derive finite difference approximations for higher-order derivatives. For example, consider the following definition for the second derivative,

 $$\displaystyle \frac{\partial ^2 U}{\partial x^2}(x)$$ $$\displaystyle =$$ $$\displaystyle \lim _{{\scriptstyle \Delta } x\rightarrow 0} \frac{{\partial U}/{\partial x}(x+{\scriptstyle \Delta } x/2) - {\partial U}/{\partial x}(x-{\scriptstyle \Delta } x/2)}{{\scriptstyle \Delta } x}$$ (2.49) $$\displaystyle =$$ $$\displaystyle \lim _{{\scriptstyle \Delta } x\rightarrow 0} \frac{1}{{\scriptstyle \Delta } x}\left[ \frac{U(x+{\scriptstyle \Delta } x)-U(x)}{{\scriptstyle \Delta } x} - \frac{U(x)-U(x-{\scriptstyle \Delta } x)}{{\scriptstyle \Delta } x}\right]$$ (2.50) $$\displaystyle =$$ $$\displaystyle \lim _{{\scriptstyle \Delta } x\rightarrow 0} \frac{1}{{\scriptstyle \Delta } x^2}\left[ U(x+{\scriptstyle \Delta } x)-2U(x) + U(x-{\scriptstyle \Delta } x) \right].$$ (2.51)

The finite difference approximation for the second order derivative is obtained eliminating the limiting process.

 $\left.\frac{\partial ^2 U}{\partial x^2}\right|_{i,j} \approx \delta _{x}^2 U_{i,j} \equiv \frac{1}{{\scriptstyle \Delta } x^2}\left(U_{i+1,j} - 2 U_{i,j} + U_{i-1,j}\right). \label{equ:uxx_ centraldiff}$ (2.52)

## Finite Difference Approximations in 2D

We can easily extend the concept of finite difference approximations to multiple spatial dimensions. In this case we represent the solution on a structured spatial mesh as shown in Figure 2.9.

Figure 2.9: Two-dimensional structured mesh for finite difference approximations.

Using $$U_{i,j}$$ to denote the value of $$U$$ at node $$(i,j)$$ our central derivative approximations for the first derivatives are:

 $$\displaystyle \left.\dfrac {\partial U}{\partial x}\right|_{i,j}$$ $$\displaystyle \approx$$ $$\displaystyle \frac{U_{i+1,j} - U_{i-1,j}}{2 \Delta x} \equiv \delta _{2x} U_{i,j}.$$ (2.53) $$\displaystyle \left.\dfrac {\partial U}{\partial y}\right|_{i,j}$$ $$\displaystyle \approx$$ $$\displaystyle \frac{U_{i,j+1} - U_{i,j-1}}{2 \Delta y} \equiv \delta _{2y} U_{i,j}.$$ (2.54)

In general, finite difference approximations will involve a set of points surrounding $$U_{i,j}$$.