FTCS Scheme

The FTCS scheme (Forward-Time Centred-Space) is an explicit finite difference method for the Heat equation.

Discretisation

Using a forward difference in time and a centred second difference in space:

$$u_{i+1,n}=(1-2r)u_{i,n}+r\left(u_{i,n+1}+u_{i,n-1}\right)$$

where the parameter $r$ is

$$r=\frac{\alpha dt}{d{x}^2}$$

Here $i$ is the time index and $n$ is the spatial index. The right-hand side contains only values at time step $i$, making this a fully explicit scheme.

Stability

The FTCS scheme is conditionally stable. The stability condition is

$$r\le \frac{1}{2}⟺dt\le \frac{d{x}^2}{2\alpha }$$

Violating this condition leads to exponentially growing oscillations. (Details of the stability proof via Fourier analysis are in Hoffman (2001) and are not covered here.)

Accuracy

• First-order in time: $O(dt)$
• Second-order in space: $O(d{x}^2)$

Python

Python

import numpy as np def ftcs_step( u_current: np.ndarray, r: float, ) -> np.ndarray: """Advance the heat equation one time step using the FTCS scheme. Args: u_current: Spatial profile at the current time step (interior points only). r: Stability parameter alpha * dt / dx**2. Must satisfy r <= 0.5. Returns: Spatial profile at the next time step. """ u_next = np.empty_like(u_current) u_next[1:-1] = (1 - 2 * r) * u_current[1:-1] + r * (u_current[2:] + u_current[:-2]) return u_next

Output

Heat equation | BTCS scheme | Finite differences | Boundary conditions | Stability of a method