We have seen that the Black and Scholes equation may be recast as the heat equation which predated the work of Black and Scholes
$$\dfrac{\partial{u}}{\partial{t}} = \dfrac{\partial^2{u}}{\partial{x^2}}$$
We also saw in the introduction to Finite Difference Methods for PDEs how we could use Taylor expansions to obtain approximate values of the first and second partial derivatives. We can now expand this to the bivariate function $u(x,t).$ This is where will see the power of numerical methods at work in dealing with two or more variables. In fact, one of the main ideas of numerical methods is to take a calculus problem and turn it into a linear algebra problem.
Observe with care that in the univariate ODE example $dy/dx$ we used small increment $h(x)$ to obtain an approximate solution. So for
$$\dfrac{\partial{u}}{\partial{t}}(x,t)$$
we are going to require small time-step increments $\Delta t$ on $t$ to obtain an approximate solution.
$$\dfrac{\partial{u}}{\partial{t}} \simeq \dfrac{u(x,t + \Delta t ) \; – \; u(x,t)}{\Delta t}, \qquad \qquad \Delta t>0 $$
Similarly, we can approximate for the second partial derivative with respect to $x$
$$\dfrac{\partial^2{u}}{\partial{x^2}} \simeq \dfrac{u(x + \Delta x, t )\; – \; 2u(x,t) \; + \; u(x – \Delta x,t)}{(\Delta x)^2}$$
Using the two difference equations we create a simple method called Explicit Euler.