Second-Order Finite Difference Scheme

The simplest, and traditional way of discretizing the 1-D wave equation is by replacing the second derivatives by second order differences:

$$\begin{eqnarray*} \left.\frac{\partial^{2} u}{\partial t^{2}}\right\vert _{x=k\Delta,t=nT} &\simeq& \frac{u_{k}^{n-1}-2u_{k}^{n}+u_{k}^{n+1}}{T^{2}} \\ \left.\frac{\partial^{2} u}{\partial x^{2}}\right\vert _{x=k\Delta,t=nT} &\simeq& \frac{u_{k-1}^{n}-2u_{k}^{n}+u_{k+1}^{n}}{\Delta^{2}} \end{eqnarray*}$$

where $u_{k}^{n}$ is defined as $u(k\Delta, nT)$ . Here we have sampled the time-space plane in a uniform grid, with a timestep of $T$ and a space step of $\Delta$ . The $u_{k}^{n}$ are the grid variables here. Now, through substitution, the wave equation becomes:

$$\begin{eqnarray*} u_{k}^{n-1}-2u_{k}^{n}+u_{k}^{n+1} = \frac{c^{2}T^{2}}{\Delta^{2}} (u_{k-1}^{n}-2u_{k}^{n}+u_{k+1}^{n}) \end{eqnarray*}$$

• Note that if we choose $T/\Delta = 1/c$ , the equation reduces further to:
  $$\begin{eqnarray*} u_{k}^{n+1} = u_{k-1}^{n}+u_{k+1}^{n}-u_{k}^{n-1} \end{eqnarray*}$$
  

  Let's examine this recursion on the time/space grid, assuming for the moment no boundary conditions: