In Computational Fluid Dynamics (CFD), solving the convection-diffusion equation is a core requirement. The steady-state 1D convection-diffusion equation is represented as:

\[\rho u \frac{d\phi}{dx} = \frac{d}{dx} \left( \Gamma \frac{d\phi}{dx} \right)\]

Where \(\rho\) is density, \(u\) is velocity, \(\phi\) is the conserved scalar, and \(\Gamma\) is the diffusion coefficient. Discretizing the convective term determines the stability and accuracy of the numerical model.

Central Differencing Scheme (CDS)

CDS uses linear interpolation to compute face values. For cell node \(P\) with neighbors \(W\) and \(E\), the convective flux at the east face \(e\) is approximated as:

\[\phi_e = \frac{\phi_P + \phi_E}{2}\]

Taylor series expansion shows CDS is second-order accurate. However, it is unconditionally unstable when the grid Peclet number exceeds 2: \(Pe_g = \frac{\rho u \Delta x}{\Gamma} > 2\). This leads to unphysical spatial oscillations (wiggles) in regions of high convective gradients.

Upwind Differencing Scheme (UDS)

To ensure physical bounds, UDS assumes that the face value matches the upstream node value: \(\phi_e = \phi_P\) if \(u > 0\), and \(\phi_e = \phi_E\) if \(u < 0\). UDS is unconditionally stable and satisfies the boundedness criterion, but introduces a large truncation error. This error acts as a artificial diffusion coefficient:

\[\Gamma_{num} = \frac{\rho u \Delta x}{2}\]

References

  • Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow. McGraw-Hill.
  • Versteeg, H. K., & Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson.