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Interactive Tool — CFD Numerics

Convection-Diffusion Equation Simulator
(Numerical Stability Visualization)

Compare steady-state solutions using central differencing, upwind differencing, and the QUICK scheme against the analytical solution. Watch how central differencing develops spurious oscillations as the Peclet number increases — an intuitive introduction to CFD numerical schemes.

$u\dfrac{d\phi}{dx} = D\dfrac{d^2\phi}{dx^2}, \quad Pe = \dfrac{u \Delta x}{D}$
Steady-State Solution Comparison Central Differencing
Numerical Characteristics Summary
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Peclet number Pe = uΔx/D
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Numerical diffusivity Dnum
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Max error |φ_num − φ_exact|

Theory Note — Convection-Diffusion & Numerical Stability

Analytical solution of the steady 1-D convection-diffusion equation (φ(0)=1, φ(L)=0):

$$\phi(x) = \frac{e^{Pe \cdot x/L} - e^{Pe}}{1 - e^{Pe}}, \quad Pe_{total} = \frac{uL}{D}$$

The central differencing scheme produces non-physical oscillations when the cell Peclet number $Pe_{cell} = u\Delta x / D > 2$. The upwind scheme is unconditionally stable but introduces numerical diffusion $D_{num} = u\Delta x/2$. QUICK offers third-order accuracy as a middle ground, though it may still exhibit mild oscillations under certain conditions.