the FTCS scheme

Imagine you know the temperature along a metal rod right now, and you want to march it forward one tiny tick of time. The simplest possible rule is: the new temperature at each point is the old one, nudged by how curved the temperature profile is around it — hot spots cool, cold dips warm. FTCS, which stands for Forward-Time Central-Space, is exactly this rule for the heat equation, and it is the canonical first explicit scheme everyone meets.

Here is the recipe for u_t = k u_xx. Use a forward difference in time, (u_new - u_old)/dt, and the central stencil in space, (u(x+h) - 2 u(x) + u(x-h))/h^2. Setting them equal and solving for the new value gives an explicit update: u_new(x) = u_old(x) + r (u_old(x+h) - 2 u_old(x) + u_old(x-h)), where r = k dt / h^2 is the mesh ratio. 'Explicit' means each new value is computed directly from old, already-known values — no system to solve, just one sweep across the grid. It is trivial to program and cheap per step.

The catch — and it is a famous one — is stability. FTCS for the heat equation is only stable when r = k dt / h^2 is at most 1/2. If you refine the space grid by halving h, you must cut the time step by a factor of four to stay below the limit, so explicit schemes become punishingly expensive on fine grids. Push past r = 1/2 and the numerical solution does not just get inaccurate, it explodes into wild oscillations of growing amplitude. This single inequality is the reason implicit schemes like backward Euler and Crank–Nicolson exist.