The Discrete Laplacian
$$\nabla^2 f(x,y) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$$
$$\nabla^2 f_{i,j} \approx \frac{f_{i-1,j}-2f_{i,j}+f_{i+1,j}}{h^2} + \frac{f_{i,j-1}-2f_{i,j}+f_{i,j+1}}{h^2}$$
$$\nabla^2 f_{i,j} \approx \frac{-4f_{i,j}+f_{i-1,j}+f_{i+1,j} + f_{i,j-1} + f_{i,j+1}}{h^2}$$
Bringing in Time
Acceleration means we need \(\frac{\partial^2 f}{\partial t^2}\)!
$$\frac {\partial f_{i,j}^n}{\partial t^2} = \frac{f_{i,j}^{n-1}-2f_{i,j}^{n}+f_{i,j}^{n+1}}{\delta t^2}$$
$$f^{new} = 2 f^{current} - f^{previous} + \delta t^2 \frac{\partial^2 f}{\partial t^2}$$