Alternative Schemes

1. Central Differencing Scheme

The central differencing scheme is a second-order accurate method used to discretize the convection-diffusion equation. It calculates fluxes at the control volume faces by averaging values from neighboring nodes. This scheme works well in diffusion-dominated flows (low Peclet number) but can become unstable in convection-dominated flows (high Peclet number) due to its inability to account for flow direction, leading to non-physical solutions.

2. Upwind Differencing Scheme

The upwind differencing scheme is a first-order accurate method that accounts for the direction of flow when calculating fluxes. It uses values from the upstream node to determine the flux at a cell face, making it more stable than the central differencing scheme, especially in high Peclet number scenarios. This scheme is widely used in computational fluid dynamics (CFD) due to its robustness, though it is less accurate than central differencing.

  • Conservativeness: The upwind differencing scheme is conservative, as it uses consistent expressions to calculate fluxes through cell faces.
  • Boundedness: This scheme ensures positive coefficients and satisfies the boundedness criteria. The matrix of coefficients is diagonally dominant, preventing oscillations in the solution.
  • Transportiveness: Unlike the central differencing scheme, upwind differencing accounts for the direction of flow. This means the convected value at a cell face is taken from the upstream node, making it more stable in high Peclet number (Pe) conditions where convection dominates.
  • Accuracy: The scheme is first-order accurate, which means it has lower accuracy compared to the central differencing scheme, which is second-order accurate. However, its robustness in handling convective flows makes it preferable in situations where the central differencing scheme fails.

3. Hybrid Differencing Scheme

The hybrid differencing scheme combines the central and upwind schemes, switching between them based on the local Peclet number. For low Peclet numbers (Pe < 2), it uses central differencing for higher accuracy, while for high Peclet numbers (Pe ≥ 2), it switches to upwind differencing to maintain stability. This scheme offers a good balance between accuracy and stability, making it suitable for a wide range of flow conditions.

  • Conservativeness: This scheme is conservative, as it uses a combination of central and upwind differencing based on the local Peclet number.
  • Boundedness: The hybrid scheme maintains positive coefficients, ensuring unconditional boundedness.
  • Transportiveness: By switching to upwind differencing for large Peclet numbers, the hybrid scheme incorporates the directionality of the flow, thus satisfying the transportiveness criterion.
  • Accuracy: While the scheme is only first-order accurate (similar to the upwind scheme), it offers a good balance between accuracy and stability. It applies central differencing for Pe < 2, thus utilizing the higher accuracy of central differencing where appropriate.

4. Power-Law Differencing Scheme

The power-law differencing scheme is a more accurate alternative to the hybrid scheme. It approximates the exact solution of the convection-diffusion equation more closely, particularly in one-dimensional problems. The scheme adjusts the flux calculation based on the Peclet number, providing better accuracy than the hybrid scheme, especially for moderate Peclet numbers (0 < Pe < 10), while maintaining stability.

  • Conservativeness: Like the hybrid scheme, the power-law differencing scheme is conservative.
  • Boundedness: It also maintains positive coefficients and is unconditionally bounded.
  • Transportiveness: The scheme accurately represents the flow characteristics by modifying the flux evaluation based on the local Peclet number, similar to the hybrid scheme.
  • Accuracy: The power-law scheme provides a closer approximation to the exact solution of the one-dimensional convection-diffusion equation than the hybrid scheme, especially for Pe < 10. Although still first-order accurate in the Taylor series truncation error, it tends to produce better results in practical applications, making it more suitable for one-dimensional problems.

Comparison with the Central Differencing Scheme

  • Stability: The central differencing scheme can become unstable when the Peclet number exceeds 2 (Pe > 2) because it does not consider flow direction, leading to negative coefficients and physically impossible solutions. In contrast, the upwind, hybrid, and power-law schemes are stable even at higher Peclet numbers due to their consideration of flow direction and use of bounded coefficients.
  • Accuracy: The central differencing scheme is second-order accurate but only remains stable and accurate for low Peclet numbers (Pe < 2). The upwind and hybrid schemes, while first-order accurate, provide stability and physically realistic solutions in a broader range of flow conditions. The power-law scheme, while still first-order accurate, offers better accuracy than the hybrid scheme for one-dimensional problems.

Conclusion

The upwind, hybrid, and power-law differencing schemes offer more stable and physically realistic solutions compared to the central differencing scheme, especially in convection-dominated flows (high Peclet numbers). Although they are less accurate in terms of truncation error, their robustness in handling practical flow problems makes them preferable in many CFD applications. The choice of scheme depends on the specific flow conditions and the need to balance accuracy with stability.