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📐 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
FTCS Heat Equation Stability Cfd
📊 Solver Telemetry ● ACTIVE
👁️ Views 49
⚡ Solves 41
💾 Downloads 382 📦 Fortran Code 4.4 KB
📅 Released Jun 2026
⏱️ Latency < 1 ms
⚡ TOOLS & REPORTS:
💾 Download Fortran 90
Thermal Systems: Copper CPU Microchannel Heat Sink Silicon Wafer Thermal Transients Forging Steel Billet Quenching (Implicit BE) Refractory Insulation Wall (Crank-Nicolson)

📥 Material Properties & Grid Spacing

📖 Heat Conduction Stability: $$\alpha = \frac{k}{\rho c_p}, \quad \text{Fo}_{\Delta} = \alpha \Delta t \left(\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} + \frac{1}{\Delta z^2}\right)$$ $$\text{FTCS Explicit Limit: } \text{Fo}_{\Delta} \le \frac{1}{2 d}, \quad \Delta t_{\text{max}} = \frac{1}{2 d\, \alpha\, \sum \Delta x_i^{-2}}$$
UNSTABLE ⚠️
Stability Status
0.4638
Grid Fourier Number ($\text{Fo}_\Delta$)
5.39e-4 s
Max Time Step ($\Delta t_{\text{max}}$)
1.16e-4
Diffusivity $\alpha$ [$\text{m}^2/\text{s}$]
⚠️ Thermal Transient Solver Diagnostic
For 2D FTCS Explicit (Forward Euler) with thermal diffusivity $\alpha = \mathbf{1.160e-4\ \text{m}^2/\text{s}}$, the grid cell diffusion time scale is $t_{\text{diff}} = \mathbf{2.16e-3\ \text{s}}$. The time step $\Delta t = 5.00e-4\ \text{s}$ exceeds the critical limit ($\Delta t_{\text{max}} = 5.39e-4\ \text{s}$). The explicit solver will produce unphysical negative temperatures and blow up. Reduce $\Delta t$ by at least -7%.

📈 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)     =   1.159555E-04 m2/s
  k (conductivity)        =   4.000000E+02 W/m.K
  rho                     =    8960.0000 kg/m3
  cp                      =     385.0000 J/kg.K
  T_init                  =       300.00 K
  T_bc                    =       500.00 K

--- GRID INFORMATION ----------------------------------------
  dx                      =   5.000000E-04 m
  dy                      =   5.000000E-04 m
  Dimensions              =  2D

--- SCHEME INFORMATION --------------------------------------
  Scheme                  = FTCS Explicit
  Stability type          = Conditionally Stable
  Fo_max (limit)          =     0.250000
  Accuracy order          = O(dt,dx2)

--- TIME STEP -----------------------------------------------
  dt (used)               =   5.000000E-04 s
  dt_max (FTCS stable)    =   2.695000E-04 s

--- FOURIER NUMBERS -----------------------------------------
  Fo (total)              =       0.463822
  rx = alpha*dt/dx2       =       0.231911
  ry = alpha*dt/dy2       =       0.231911
  Fo_max (scheme)         =       0.250000

--- STABILITY STATUS ----------------------------------------
  STATUS                  = UNSTABLE
  Margin                  =       -85.53 %

--- DIFFUSION CHARACTERISTICS -------------------------------
  t_cell = dx2/alpha      =   2.156000E-03 s
  Penetration depth       =   4.815713E-04 m
  Pen. depth / dx         =         0.9631
  t_diff (10*dx domain)   =   2.156000E-01 s
  Steps to t_diff         =          431.2

--- ALL SCHEMES COMPARISON ----------------------------------
  Scheme              Fo_max      Stable?  Order
  --------------------------------------------------------
  FTCS                   0.2500      NO       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
  --------------------------------------------------------------------------
  1.3475E-05    0.012500    0.006250  STABLE      7.9057E-05    1.6000E+04
  4.0771E-05    0.037821    0.018910  STABLE      1.3751E-04    5.2881E+03
  6.8066E-05    0.063141    0.031571  STABLE      1.7768E-04    3.1675E+03
  9.5362E-05    0.088462    0.044231  STABLE      2.1031E-04    2.2609E+03
  1.2266E-04    0.113782    0.056891  STABLE      2.3852E-04    1.7577E+03
  1.4995E-04    0.139103    0.069551  STABLE      2.6373E-04    1.4378E+03
  1.7725E-04    0.164423    0.082212  STABLE      2.8673E-04    1.2164E+03
  2.0454E-04    0.189744    0.094872  STABLE      3.0801E-04    1.0541E+03
  2.3184E-04    0.215064    0.107532  STABLE      3.2792E-04    9.2996E+02
  2.5913E-04    0.240385    0.120192  STABLE      3.4669E-04    8.3200E+02
  2.8643E-04    0.265705    0.132853  UNSTABLE    3.6449E-04    7.5271E+02
  3.1373E-04    0.291026    0.145513  UNSTABLE    3.8146E-04    6.8722E+02
  3.4102E-04    0.316346    0.158173  UNSTABLE    3.9771E-04    6.3222E+02
  3.6832E-04    0.341667    0.170833  UNSTABLE    4.1332E-04    5.8537E+02
  3.9561E-04    0.366987    0.183494  UNSTABLE    4.2836E-04    5.4498E+02
  4.2291E-04    0.392308    0.196154  UNSTABLE    4.4289E-04    5.0980E+02
  4.5020E-04    0.417628    0.208814  UNSTABLE    4.5696E-04    4.7889E+02
  4.7750E-04    0.442949    0.221474  UNSTABLE    4.7061E-04    4.5152E+02
  5.0479E-04    0.468269    0.234135  UNSTABLE    4.8387E-04    4.2710E+02
  5.3209E-04    0.493590    0.246795  UNSTABLE    4.9678E-04    4.0519E+02
  5.5939E-04    0.518910    0.259455  UNSTABLE    5.0937E-04    3.8542E+02
  5.8668E-04    0.544231    0.272115  UNSTABLE    5.2165E-04    3.6749E+02
  6.1398E-04    0.569551    0.284776  UNSTABLE    5.3364E-04    3.5115E+02
  6.4127E-04    0.594872    0.297436  UNSTABLE    5.4538E-04    3.3621E+02
  6.6857E-04    0.620192    0.310096  UNSTABLE    5.5686E-04    3.2248E+02
  6.9586E-04    0.645513    0.322756  UNSTABLE    5.6812E-04    3.0983E+02
  7.2316E-04    0.670833    0.335417  UNSTABLE    5.7915E-04    2.9814E+02
  7.5045E-04    0.696154    0.348077  UNSTABLE    5.8998E-04    2.8729E+02
  7.7775E-04    0.721474    0.360737  UNSTABLE    6.0061E-04    2.7721E+02
  8.0504E-04    0.746795    0.373397  UNSTABLE    6.1106E-04    2.6781E+02
  8.3234E-04    0.772115    0.386058  UNSTABLE    6.2134E-04    2.5903E+02
  8.5964E-04    0.797436    0.398718  UNSTABLE    6.3144E-04    2.5080E+02
  8.8693E-04    0.822756    0.411378  UNSTABLE    6.4139E-04    2.4309E+02
  9.1423E-04    0.848077    0.424038  UNSTABLE    6.5118E-04    2.3583E+02
  9.4152E-04    0.873397    0.436699  UNSTABLE    6.6083E-04    2.2899E+02
  9.6882E-04    0.898718    0.449359  UNSTABLE    6.7034E-04    2.2254E+02
  9.9611E-04    0.924038    0.462019  UNSTABLE    6.7972E-04    2.1644E+02
  1.0234E-03    0.949359    0.474679  UNSTABLE    6.8897E-04    2.1067E+02
  1.0507E-03    0.974679    0.487340  UNSTABLE    6.9810E-04    2.0520E+02
  1.0780E-03    1.000000    0.500000  UNSTABLE    7.0711E-04    2.0000E+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)
============================================================
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📘 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

  1. Incropera, F. P. et al.: Fundamentals of Heat and Mass Transfer, Ch. 5.
  2. 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 $\mathbf{11.9\text{ ms}}$.