📐 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
0.0052
Grid Fourier Number ($\text{Fo}_\Delta$)
Unbounded
Max Time Step ($\Delta t_{\text{max}}$)
4.17e-7
Diffusivity $\alpha$ [$\text{m}^2/\text{s}$]
💡
Thermal Transient Solver Diagnostic
For 1D Crank-Nicolson (Implicit Trapezoidal) with thermal diffusivity $\alpha = \mathbf{4.167e-7\ \text{m}^2/\text{s}}$, the grid cell diffusion time scale is $t_{\text{diff}} = \mathbf{9.60e+2\ \text{s}}$.
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: $\text{Fo}_{\text{crit}} = 0.500$🔍 View Raw GNU Fortran Double-Precision Solver Output
============================================================ HEAT EQUATION STABILITY CALCULATOR ============================================================ --- INPUT CONDITIONS ---------------------------------------- alpha (diffusivity) = 4.166667E-07 m2/s k (conductivity) = 2.500000E-01 W/m.K rho = 600.0000 kg/m3 cp = 1000.0000 J/kg.K T_init = 300.00 K T_bc = 500.00 K --- GRID INFORMATION ---------------------------------------- dx = 2.000000E-02 m Dimensions = 1D --- SCHEME INFORMATION -------------------------------------- Scheme = Crank-Nicolson Stability type = Unconditionally Stable Fo_max (limit) = unlimited Accuracy order = O(dt2,dx2) --- TIME STEP ----------------------------------------------- dt (used) = 5.000000E+00 s dt_max (FTCS stable) = 1.000000E+10 s --- FOURIER NUMBERS ----------------------------------------- Fo (total) = 0.005208 rx = alpha*dt/dx2 = 0.005208 --- STABILITY STATUS ---------------------------------------- STATUS = UNCONDITIONAL Margin = 100.00 % --- DIFFUSION CHARACTERISTICS ------------------------------- t_cell = dx2/alpha = 9.600000E+02 s Penetration depth = 2.886751E-03 m Pen. depth / dx = 0.1443 t_diff (10*dx domain) = 9.600000E+04 s Steps to t_diff = 19200.0 --- ALL SCHEMES COMPARISON ---------------------------------- Scheme Fo_max Stable? Order -------------------------------------------------------- FTCS 0.5000 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-01 0.000260 0.000260 STABLE 6.4550E-04 3.8400E+05 5.3718E+00 0.005596 0.005596 STABLE 2.9922E-03 1.7871E+04 1.0494E+01 0.010931 0.010931 STABLE 4.1820E-03 9.1484E+03 1.5615E+01 0.016266 0.016266 STABLE 5.1015E-03 6.1478E+03 2.0737E+01 0.021601 0.021601 STABLE 5.8789E-03 4.6294E+03 2.5859E+01 0.026936 0.026936 STABLE 6.5649E-03 3.7124E+03 3.0981E+01 0.032272 0.032272 STABLE 7.1857E-03 3.0987E+03 3.6103E+01 0.037607 0.037607 STABLE 7.7570E-03 2.6591E+03 4.1224E+01 0.042942 0.042942 STABLE 8.2890E-03 2.3287E+03 4.6346E+01 0.048277 0.048277 STABLE 8.7888E-03 2.0714E+03 5.1468E+01 0.053612 0.053612 STABLE 9.2617E-03 1.8652E+03 5.6590E+01 0.058948 0.058948 STABLE 9.7117E-03 1.6964E+03 6.1712E+01 0.064283 0.064283 STABLE 1.0142E-02 1.5556E+03 6.6833E+01 0.069618 0.069618 STABLE 1.0554E-02 1.4364E+03 7.1955E+01 0.074953 0.074953 STABLE 1.0951E-02 1.3342E+03 7.7077E+01 0.080288 0.080288 STABLE 1.1334E-02 1.2455E+03 8.2199E+01 0.085624 0.085624 STABLE 1.1705E-02 1.1679E+03 8.7321E+01 0.090959 0.090959 STABLE 1.2064E-02 1.0994E+03 9.2442E+01 0.096294 0.096294 STABLE 1.2413E-02 1.0385E+03 9.7564E+01 0.101629 0.101629 STABLE 1.2752E-02 9.8397E+02 1.0269E+02 0.106964 0.106964 STABLE 1.3082E-02 9.3489E+02 1.0781E+02 0.112300 0.112300 STABLE 1.3404E-02 8.9047E+02 1.1293E+02 0.117635 0.117635 STABLE 1.3719E-02 8.5009E+02 1.1805E+02 0.122970 0.122970 STABLE 1.4027E-02 8.1321E+02 1.2317E+02 0.128305 0.128305 STABLE 1.4328E-02 7.7939E+02 1.2829E+02 0.133640 0.133640 STABLE 1.4623E-02 7.4828E+02 1.3342E+02 0.138976 0.138976 STABLE 1.4912E-02 7.1955E+02 1.3854E+02 0.144311 0.144311 STABLE 1.5195E-02 6.9295E+02 1.4366E+02 0.149646 0.149646 STABLE 1.5474E-02 6.6824E+02 1.4878E+02 0.154981 0.154981 STABLE 1.5747E-02 6.4524E+02 1.5390E+02 0.160317 0.160317 STABLE 1.6016E-02 6.2377E+02 1.5903E+02 0.165652 0.165652 STABLE 1.6280E-02 6.0368E+02 1.6415E+02 0.170987 0.170987 STABLE 1.6540E-02 5.8484E+02 1.6927E+02 0.176322 0.176322 STABLE 1.6796E-02 5.6714E+02 1.7439E+02 0.181657 0.181657 STABLE 1.7049E-02 5.5049E+02 1.7951E+02 0.186993 0.186993 STABLE 1.7297E-02 5.3478E+02 1.8463E+02 0.192328 0.192328 STABLE 1.7542E-02 5.1995E+02 1.8976E+02 0.197663 0.197663 STABLE 1.7784E-02 5.0591E+02 1.9488E+02 0.202998 0.202998 STABLE 1.8022E-02 4.9262E+02 2.0000E+02 0.208333 0.208333 STABLE 1.8257E-02 4.8000E+02 --- 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 $\mathbf{11.9\text{ ms}}$.