🔥 Heat Equation Stability — Fourier Number
Fo = αΔt/Δx² ≤ 1/2. Compare FTCS, Implicit, CN, DuFort-Frankel. Ref: §2.7.2, Eq.2.80.
🔥 Heat Diffusion
Configuration
$\text{Fo}=\dfrac{\alpha\Delta t}{\Delta x^2}\le\dfrac{1}{2}$
$\Delta t_{\max}=\dfrac{\Delta x^2}{2\alpha}$
$\Delta t_{\max}=\dfrac{\Delta x^2}{2\alpha}$
Results
Click Check Stability.
About Fo: For FTCS, Fo ≤ 1/(2·n_dim). Implicit/CN are unconditionally stable.
Methodology
Theory
$$\text{Fo}=\frac{\alpha\Delta t}{\Delta x^2}\le\frac{1}{2}\;\text{(1D)}$$ $$\text{2D: Fo}\le\frac{1}{4},\;\text{3D: Fo}\le\frac{1}{6}$$ $$\delta_p=2\sqrt{\alpha t}$$
Example
Steel, α=1.17e-5, Δx=1mm
dt_max=0.0427s
Fo=0.45 ✅
dt_max=0.0427s
Fo=0.45 ✅
References
- Gupta §2.7.2 Eq.2.80
- Incropera, Heat Transfer
- Patankar, Num. HT
- Crank & Nicolson, 1947