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Explicit Euler Stability for the Heat Equation

Article Quant Q&A · Author: Victor
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Summary

The document asks why the explicit finite-difference scheme for the heat equation requires a relationship between its time step and spatial grid size. It gives the forward-time, centered-space update equation, where k is the time mesh size and h is the space mesh size, and points to the stability condition k ≤ h²/2.

The response emphasizes that the time and space steps cannot be selected independently if the numerical solution is to remain stable. It directs readers to a textbook discussion and a linked short explanation, but does not derive the condition, provide a stability analysis, or show numerical evidence. The note is therefore a concise statement of a stability constraint rather than a worked tutorial. Its scope is limited to the stated explicit Euler discretization of the heat equation; it does not discuss alternative schemes or broader applications to finance.

Key ideas

  • The explicit Euler discretization uses a forward difference in time and a centered second difference in space.
  • Stability requires the time step to satisfy k ≤ h²/2.
  • The time and space mesh sizes must be chosen together under this condition.
  • The document states the condition but does not show its derivation.

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# Explicit Euler stability for the Heat Equation (FDM)


# Explicit Euler stability for the Heat Equation (FDM)












Why the Explicit Euler scheme for the Heat Equation is stable only if $k \leq h^2/2$ ?

Here is the difference equation: \begin{equation} \frac{U_j^{n+1}-U_{j}^n}{k} = \frac{1}{h^2}(U_{j+1}^n-2U_j^n+U_{j-1}^n) \end{equation} with $k$ the mesh size in time and $h$ the mesh size in space

## Answer by FunnyBuzer (score 3)

https://quant.stackexchange.com/a/44302

You can consult Seydel pages 99-106 for explicit FD or for a short summary this link. The idea is that you cannot choose $k$ and $\frac{h^2}{2}$ independently for stability.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.