the computational grid

A partial differential equation lives on a continuum — temperature is defined at every point of a metal bar and at every instant of time, an uncountable infinity of values. A computer cannot store infinitely many numbers, so the first move in solving any PDE by finite differences is to give up the continuum and keep only a finite scaffolding of sample points: the computational grid. Picture graph paper laid over the bar, with the temperature tracked only at the crossings of the lines, and only at a ticking sequence of time instants.

Concretely, you chop space into equal steps of size h (often written delta-x) and time into equal steps of size k (often delta-t). On a 1D bar of length L you keep the points x_j = j * h for j = 0, 1, ..., N, and on time you keep t_n = n * k. The unknown becomes a table of numbers u_j^n, meant to approximate the true solution u(x_j, t_n) at grid point j and time level n. In two space dimensions the grid is a rectangular array of nodes (x_i, y_j) and the unknown is u_{i,j}. The whole PDE problem is thus replaced by the problem of filling in this table consistently with discretized versions of the equation.

The grid is where every other choice in finite differences plays out: the spacing h controls how much detail you can resolve and how small the truncation error is (typically O(h^2) for a good scheme), while the time step k is constrained by stability (the CFL condition for explicit schemes). Finer grids are more accurate but cost more memory and arithmetic — in 2D, halving h quadruples the number of nodes, and in 3D it multiplies them by eight, the reason large simulations are so expensive. Choosing the grid is not a formality; it is where accuracy, cost, and stability all collide.