Numerical Stability and Accuracy

The central differencing scheme is a popular method used for solving convection-diffusion problems due to its simplicity and second-order accuracy. However, its numerical stability and accuracy depend heavily on the conditions under which it is applied. Here’s a discussion based on the information provided:

Numerical Stability

The numerical stability of the central differencing scheme is influenced by the Peclet number (Pe), which is a dimensionless number representing the ratio of convection to diffusion in a flow system. Specifically, the stability of the scheme is contingent on the following:

  1. Boundedness and the Peclet Number (Pe < 2): Find more on Peclet Number
    • The scheme is stable and produces physically realistic solutions when the Peclet number is less than 2. This condition ensures that the convective contribution to the coefficients remains positive, preventing non-physical behavior like oscillations or negative coefficients.
    • When the Peclet number exceeds 2, the scheme becomes unbounded, potentially leading to physically impossible solutions such as large undershoots or overshoots in the computed variable. This instability arises because the central differencing scheme does not inherently account for the direction of the flow or the relative strength of convection versus diffusion.
  2. Impact of Flow Direction and Convection Strength:
    • The central differencing scheme introduces influence from all neighboring nodes when computing convective and diffusive fluxes. It fails to properly recognize the direction of the flow, which can lead to inaccuracies when convection dominates diffusion (i.e., at high Peclet numbers).
    • In scenarios where convection is strong relative to diffusion, the scheme may not maintain stability due to its inability to properly transport the properties in the direction of the flow.

Accuracy

The accuracy of the central differencing scheme is characterized by its truncation error, which is second-order. This implies that the error in the approximation decreases quadratically as the grid spacing decreases, making it a relatively accurate scheme under the right conditions. However, its accuracy is also influenced by the Peclet number:

  1. Accuracy at Low Peclet Numbers (Pe < 2):

    • The scheme is accurate when the Peclet number is low, corresponding to diffusion-dominated flows or flows with small velocities and/or small grid spacing. In such conditions, the central differencing scheme can provide solutions that are close to the analytical solution.

  2. Decreased Accuracy at High Peclet Numbers:

    • When the Peclet number exceeds 2, the scheme’s accuracy diminishes significantly, as it no longer properly represents the balance between convection and diffusion. This can lead to errors such as non-physical oscillations and incorrect property distributions.

Conditions Leading to Instability

The central differencing scheme may become unstable under the following conditions:

  • High Peclet Number (Pe > 2): When convection dominates diffusion, leading to large Peclet numbers, the scheme can become unstable due to negative coefficients, resulting in unbounded solutions.

  • Inadequate Grid Resolution: If the grid spacing is not sufficiently small in regions with high velocity gradients, the scheme may fail to accurately capture the physical behavior of the flow, leading to instability.

  • Strong Convective Flows: In scenarios where the convective effects are much stronger than diffusive effects, the central differencing scheme may not provide accurate or stable results due to its inherent inability to properly account for the directionality of flow.

Summary

The central differencing scheme is stable and accurate for convection-diffusion problems when the Peclet number is low (Pe < 2), corresponding to diffusion-dominated flows or finely resolved grids. However, it can become unstable and inaccurate when the Peclet number exceeds 2, especially in cases where convection dominates. This limitation necessitates the use of alternative schemes, such as upwind or hybrid schemes, for general-purpose flow calculations, especially in convective-dominated flows.