📐 FTCS Heat Equation Stability
Analyze numerical stability limits for 1D/2D transient diffusion solvers using the FTCS scheme.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📥 Material Properties & Grid Spacing
📖 Heat Conduction Stability:
$$\alpha = \frac{k}{\rho c_p}, \quad \text{Fo}_{\Delta} = \alpha \Delta t \left(\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} + \frac{1}{\Delta z^2}\right)$$
$$\text{FTCS Explicit Limit: } \text{Fo}_{\Delta} \le \frac{1}{2 d}, \quad \Delta t_{\text{max}} = \frac{1}{2 d\, \alpha\, \sum \Delta x_i^{-2}}$$
STABLE ✅
Stability Status
0.1194
Grid Fourier Number ($\text{Fo}_\Delta$)
Unbounded
Max Time Step ($\Delta t_{\text{max}}$)
1.19e-5
Diffusivity $\alpha$ [$\text{m}^2/\text{s}$]
💡
Thermal Transient Solver Diagnostic
For 2D Implicit Backward Euler (BTCS) with thermal diffusivity $\alpha = \mathbf{1.194e-5\ \text{m}^2/\text{s}}$, the grid cell diffusion time scale is $t_{\text{diff}} = \mathbf{8.37e+0\ \text{s}}$.
The selected scheme is unconditionally stable. However, for physical accuracy, $\Delta t$ should not exceed $t_{\text{diff}}$ to avoid spatial smearing.
📈 Grid Fourier Number $\text{Fo}_\Delta$ vs Time Step $\Delta t$
Dashed Red: $\text{Fo}_{\text{crit}} = 0.250$🔍 View Raw GNU Fortran Double-Precision Solver Output
============================================================ HEAT EQUATION STABILITY CALCULATOR ============================================================ --- INPUT CONDITIONS ---------------------------------------- alpha (diffusivity) = 1.194268E-05 m2/s k (conductivity) = 4.500000E+01 W/m.K rho = 7850.0000 kg/m3 cp = 480.0000 J/kg.K T_init = 300.00 K T_bc = 500.00 K --- GRID INFORMATION ---------------------------------------- dx = 1.000000E-02 m dy = 1.000000E-02 m Dimensions = 2D --- SCHEME INFORMATION -------------------------------------- Scheme = Implicit BE Stability type = Unconditionally Stable Fo_max (limit) = unlimited Accuracy order = O(dt,dx2) --- TIME STEP ----------------------------------------------- dt (used) = 5.000000E-01 s dt_max (FTCS stable) = 1.000000E+10 s --- FOURIER NUMBERS ----------------------------------------- Fo (total) = 0.119427 rx = alpha*dt/dx2 = 0.059713 ry = alpha*dt/dy2 = 0.059713 --- STABILITY STATUS ---------------------------------------- STATUS = UNCONDITIONAL Margin = 100.00 % --- DIFFUSION CHARACTERISTICS ------------------------------- t_cell = dx2/alpha = 8.373333E+00 s Penetration depth = 4.887264E-03 m Pen. depth / dx = 0.4887 t_diff (10*dx domain) = 8.373333E+02 s Steps to t_diff = 1674.7 --- ALL SCHEMES COMPARISON ---------------------------------- Scheme Fo_max Stable? Order -------------------------------------------------------- FTCS 0.2500 YES O(dt,dx2) Implicit BE unlimited YES O(dt,dx2) Crank-Nicolson unlimited YES O(dt2,dx2) DuFort-Frankel unlimited YES* O(dt2,dx2) --- PROFILE vs dt ------------------------------------------ dt Fo rx Status pen_depth N_steps -------------------------------------------------------------------------- 2.5000E-02 0.005971 0.002986 STABLE 1.0928E-03 3.3493E+04 5.3718E-01 0.128307 0.064154 STABLE 5.0657E-03 1.5588E+03 1.0494E+00 0.250643 0.125322 UNSTABLE 7.0802E-03 7.9795E+02 1.5615E+00 0.372979 0.186489 UNSTABLE 8.6369E-03 5.3622E+02 2.0737E+00 0.495315 0.247657 UNSTABLE 9.9530E-03 4.0378E+02 2.5859E+00 0.617651 0.308825 UNSTABLE 1.1114E-02 3.2381E+02 3.0981E+00 0.739987 0.369993 UNSTABLE 1.2165E-02 2.7028E+02 3.6103E+00 0.862322 0.431161 UNSTABLE 1.3133E-02 2.3193E+02 4.1224E+00 0.984658 0.492329 UNSTABLE 1.4033E-02 2.0312E+02 4.6346E+00 1.106994 0.553497 UNSTABLE 1.4879E-02 1.8067E+02 5.1468E+00 1.229330 0.614665 UNSTABLE 1.5680E-02 1.6269E+02 5.6590E+00 1.351666 0.675833 UNSTABLE 1.6442E-02 1.4797E+02 6.1712E+00 1.474002 0.737001 UNSTABLE 1.7170E-02 1.3569E+02 6.6833E+00 1.596338 0.798169 UNSTABLE 1.7868E-02 1.2529E+02 7.1955E+00 1.718673 0.859337 UNSTABLE 1.8540E-02 1.1637E+02 7.7077E+00 1.841009 0.920505 UNSTABLE 1.9189E-02 1.0864E+02 8.2199E+00 1.963345 0.981673 UNSTABLE 1.9816E-02 1.0187E+02 8.7321E+00 2.085681 1.042841 UNSTABLE 2.0424E-02 9.5892E+01 9.2442E+00 2.208017 1.104008 UNSTABLE 2.1014E-02 9.0579E+01 9.7564E+00 2.330353 1.165176 UNSTABLE 2.1589E-02 8.5824E+01 1.0269E+01 2.452689 1.226344 UNSTABLE 2.2148E-02 8.1543E+01 1.0781E+01 2.575024 1.287512 UNSTABLE 2.2694E-02 7.7669E+01 1.1293E+01 2.697360 1.348680 UNSTABLE 2.3227E-02 7.4147E+01 1.1805E+01 2.819696 1.409848 UNSTABLE 2.3747E-02 7.0930E+01 1.2317E+01 2.942032 1.471016 UNSTABLE 2.4257E-02 6.7980E+01 1.2829E+01 3.064368 1.532184 UNSTABLE 2.4756E-02 6.5266E+01 1.3342E+01 3.186704 1.593352 UNSTABLE 2.5246E-02 6.2761E+01 1.3854E+01 3.309040 1.654520 UNSTABLE 2.5726E-02 6.0440E+01 1.4366E+01 3.431376 1.715688 UNSTABLE 2.6197E-02 5.8286E+01 1.4878E+01 3.553711 1.776856 UNSTABLE 2.6660E-02 5.6279E+01 1.5390E+01 3.676047 1.838024 UNSTABLE 2.7115E-02 5.4406E+01 1.5903E+01 3.798383 1.899192 UNSTABLE 2.7562E-02 5.2654E+01 1.6415E+01 3.920719 1.960360 UNSTABLE 2.8003E-02 5.1011E+01 1.6927E+01 4.043055 2.021527 UNSTABLE 2.8436E-02 4.9468E+01 1.7439E+01 4.165391 2.082695 UNSTABLE 2.8863E-02 4.8015E+01 1.7951E+01 4.287727 2.143863 UNSTABLE 2.9284E-02 4.6645E+01 1.8463E+01 4.410062 2.205031 UNSTABLE 2.9699E-02 4.5351E+01 1.8976E+01 4.532398 2.266199 UNSTABLE 3.0108E-02 4.4127E+01 1.9488E+01 4.654734 2.327367 UNSTABLE 3.0511E-02 4.2967E+01 2.0000E+01 4.777070 2.388535 UNSTABLE 3.0910E-02 4.1867E+01 --- EQUATIONS USED ------------------------------------------ Fo = alpha*dt/dx^2 (Fourier number, Eq. 2.80) FTCS: Fo <= 1/(2*ndim) 1D: Fo<=0.5, 2D: Fo<=0.25, 3D: Fo<=1/6 dt_max = Fo_max / (alpha * sum(1/dxi^2)) Penetration depth = 2*sqrt(alpha*t) ============================================================
📘 Calculation Methodology: Heat Equation Explicit Stability & Fourier Limit
Mathematical Model & Theory
Explicit finite difference integration of transient diffusion requires the mesh Fourier number $Fo$ to satisfy the maximum principle to avoid non-physical oscillations:
$$Fo = \frac{\alpha \Delta t}{\Delta x^2} \le \frac{1}{2 d} \quad (d = \text{spatial dimensions})$$
$$\text{1D: } Fo \le 0.5, \quad \text{2D: } Fo \le 0.25, \quad \text{3D: } Fo \le 0.167$$
Assumptions
- Explicit Forward-Time Central-Space (FTCS) finite differencing.
- Constant thermal diffusivity $lpha = k/( ho c_p)$.
Academic References
- Incropera, F. P. et al.: Fundamentals of Heat and Mass Transfer, Ch. 5.
- Patankar, S. V.: Numerical Heat Transfer, CRC Press.
Worked Engineering Example
Problem Statement:
An aluminium plate ($\alpha = 8.4 \times 10^{-5}\text{ m}^2/\text{s}$) has 2D grid spacing $\Delta x = 2.0\text{ mm}$. Calculate maximum stable time step.
Step-by-step Solution:
1. $\Delta t_{max} = 0.25 \times (0.002)^2 / (8.4 \times 10^{-5}) = 1.0 \times 10^{-6} / (8.4 \times 10^{-5}) \approx 0.0119\text{ s}$.
Final Result:
Maximum stable time step is $\mathbf{11.9\text{ ms}}$.
An aluminium plate ($\alpha = 8.4 \times 10^{-5}\text{ m}^2/\text{s}$) has 2D grid spacing $\Delta x = 2.0\text{ mm}$. Calculate maximum stable time step.
Step-by-step Solution:
1. $\Delta t_{max} = 0.25 \times (0.002)^2 / (8.4 \times 10^{-5}) = 1.0 \times 10^{-6} / (8.4 \times 10^{-5}) \approx 0.0119\text{ s}$.
Final Result:
Maximum stable time step is $\mathbf{11.9\text{ ms}}$.