Finite difference method

Consider first a one-dimensional PDE for the unknown function $u(x,t)$ . One way to numerically solve the PDE is to approximate all the derivatives by finite differences. We partition the domain in space using a mesh $x_0, x_1,\ldots,x_N$ and in time using a mesh $t_0, t_1,\ldots,t_T$ . We assume first a uniform partition both in space and in time, so the difference between two consecutive space points will be $\triangle x$ and between two consecutive time points will be $\triangle t$ , i.e.,

$$x_i = x_0+i\triangle x,\qquad i=0,1,\ldots,M;$$

$$t_j = t_0+j\triangle t,\qquad j=0,1,\ldots,T;$$