Houjun Liu

Finite Difference Method

The Finite Difference Method is a method of solving partial Differential Equations. It follows two steps:

  1. Develop discrete difference equations for the desired expression
  2. Algebraically solve these equations to yield stepped solutions


Follow Along

We will try to solve:

\begin{equation} \pdv{p(t,x)}{t} = \frac{1}{2}\pdv[2]{p(t,x)}{x} \end{equation}

To aid in notation, let us:

\begin{equation} p(t_{i}, x_{j}) := p_{i,j} \end{equation}

to represent one distinct value of our function \(p\).

Let’s begin by writing our expression above via our new notation:

\begin{equation} \pdv{p_{i,j}}{t}= \frac{1}{2} \pdv[2]{p_{i,j}}{x} \end{equation}

Great. Now, let’s think about the left side and try to turn it into a difference eqn:

What exactly is—

\begin{equation} \pdv{p_{i,j}}{t} \end{equation}

as a finite difference? Well, it is just:

\begin{equation} \frac{p_{i+1,j}-p_{i,{j}}}{\Delta t} \end{equation}

What about second partials?

Well, what is—

\begin{equation} \pdv[2]{p_{i,j}}{x} \end{equation}

It is:

\begin{equation} \frac{\pdv{p_{i,j+1}}{x}- \pdv{p_{i,j}}{x}}{\Delta x} \end{equation}

Expanding the top expressions even more difference expressions:

\begin{equation} \frac{\frac{p_{i,{j+2}}-p_{i,{j+1}}}{\Delta x}- \frac{p_{i,{j+1}}-p_{i,{j}}}{\Delta x}}{\Delta x} \end{equation}

This equals to:

\begin{equation} \frac{\frac{p_{i,{j+2}}-p_{i,{j+1}} - p_{i,{j+1}}+p_{i,{j}}}{(\Delta x)^{2}} \end{equation}

Finally, substitute this into our expression, then solve for some \(p_{{i+1}, j}\) in terms of \(p_{i, ?}\). We will treat the entire “row” of \(p_{i,?}\) as our initial condition, then solve for the rest + propagate forward.