1 Modelling context

A numerical scheme is useful only if it produces a bounded solution. A scheme is stable for a given problem if small perturbations in the initial data do not grow without bound as the integration proceeds. An unstable scheme produces a numerical solution that blows up, regardless of how accurate the underlying discretization might be on paper.

The FTCS scheme is conditionally stable: it is stable if and only if the diffusion number satisfies r ≤ 1/2. This is the CFL condition for the explicit heat equation, named for Courant, Friedrichs, and Lewy, who derived analogous conditions for hyperbolic problems.

The condition is not a technicality. It is a fundamental constraint that determines the cost of the simulation, and it explains why explicit schemes are impractical for fine spatial grids.

2 Learning objectives

3 Prerequisite review

Stability

A property of a numerical scheme applied to a particular problem: small perturbations in the data do not grow without bound as the integration proceeds.

Fourier mode

A solution of the form u_i^n = G^n e^{i k x_i}, where k is a wavenumber and G is the amplification factor. Fourier modes are the building blocks of any smooth initial condition.

Amplification factor

The complex number G(k) such that a Fourier mode with wavenumber k is multiplied by G at each time step. Stability requires |G(k)| ≤ 1 for all k.

Consistency

A scheme is consistent with a PDE if its truncation error tends to zero as the grid spacings tend to zero. FTCS is consistent.

Lax equivalence theorem

For a consistent scheme applied to a well-posed linear problem, stability is equivalent to convergence.

4 Mathematical formulation

Von Neumann stability analysis

The von Neumann method analyses the growth of Fourier modes in the numerical solution. It is the standard tool for deriving stability conditions for linear schemes with constant coefficients.

Consider the FTCS update rule from Lesson 1:

u_i^{n+1} = r u_{i−1}^n + ( 1 − 2r ) u_i^n + r u_{i+1}^n

Substitute a Fourier mode of the form u_i^n = G^n e^{i k i Δx}:

G^{n+1} e^{i k i Δx} = r G^n e^{i k (i−1) Δx} + (1 − 2r) G^n e^{i k i Δx} + r G^n e^{i k (i+1) Δx}

Dividing by G^n e^{i k i Δx}:

G = r e^{−i k Δx} + ( 1 − 2r ) + r e^{i k Δx}

Using e^{iθ} + e^{−iθ} = 2 cos θ:

G = 1 − 2r + 2r cos( k Δx )
  = 1 − 2r ( 1 − cos( k Δx ) )

The amplification factor G is real, and depends on the wavenumber through the product k Δx. Since 1 − cos θ ∈ [0, 2], the factor ranges over

G ∈ [ 1 − 4r, 1 ]

For stability, we require |G| ≤ 1 for all wavenumbers. Since G ≤ 1 automatically, the binding constraint is G ≥ −1:

1 − 4r ≥ −1
4r ≤ 2
r ≤ 1/2

This is the CFL condition for the FTCS scheme:

r = α Δt / Δx² ≤ 1/2

Equivalently,

Δt ≤ Δx² / ( 2 α )

The most unstable mode

The stability limit is set by the Fourier mode with k Δx = π, which corresponds to the highest wavenumber representable on the grid — the alternating mode that changes sign at every grid point:

cos(π) = −1  ⟹  G = 1 − 2r (1 − (−1)) = 1 − 4r

This mode is the most unstable. When r > 1/2, the amplification factor for this mode has absolute value greater than 1, and the mode grows by a factor of 4r − 1 at each time step. The signature of instability is a numerical solution dominated by alternating-sign oscillations, growing in magnitude.

Why the condition cannot be avoided

The CFL condition is a property of the explicit scheme, not of the PDE. Its origin is the coupling between the spatial and temporal discretizations: the explicit update rule at each grid point combines information from neighbouring points, and the ratio of the time step to the spatial step determines how far that information can travel within a single step.

Implicit schemes avoid the restriction by coupling all grid points at the new time level, requiring the solution of a linear system at each step. The additional cost per step is usually far less than the savings from the reduced number of steps, especially for fine grids. Implicit schemes for the heat equation are unconditionally stable.

Higher-order explicit schemes have larger stability regions but remain conditionally stable. No explicit scheme for the heat equation is unconditionally stable.

5 Worked example

∂u/∂t = 0.1 ∂²u/∂x²,    u(x, 0) = sin(π x),    u(0, t) = u(1, t) = 0

6 Verification check

7 Interpretation

The CFL condition r ≤ 1/2 has three implications that every user of the FTCS scheme should understand.

• The time step is bounded by the spatial step. Refining the spatial grid forces a reduction in the time step. The two cannot be chosen independently.
• The bound is quadratic in Δx. Halving the spatial grid requires quartering the time step. The total computational work scales as 1/Δx³, which is why explicit schemes become impractical for very fine grids.
• The instability is self-diagnosing. Violating the CFL condition produces a distinctive, unmistakable growth pattern. It is not a subtle error.

8 Uncertainty and limitations

The von Neumann stability analysis is a powerful tool, but it has limitations.

For the FTCS scheme applied to the constant-coefficient heat equation with periodic boundary conditions, the von Neumann condition r ≤ 1/2 is both necessary and sufficient. For other problems, it should be treated as a necessary condition, and additional testing may be required to confirm stability.

9 Common implementation errors

10 Independent modelling task

11 Reflection prompt

12 Lesson summary

§ Self-quiz, practice, and homework