Central Differencing Scheme for diffusion terms

The central differencing scheme is a numerical method used to approximate the diffusive terms in the convection-diffusion equation. It works by approximating the gradient of the transported property (e.g., temperature, concentration) between adjacent grid points using a linear interpolation.

In the context of diffusion, which is represented by the second derivative in the equation, the central differencing scheme assumes that the property varies linearly between neighboring grid points. This linear interpolation allows for the property values at the faces of a control volume to be approximated as the average of the values at the adjacent grid points.

The key idea behind this approach is that it leverages the average values of the property at two neighboring nodes to estimate the property at the interface (or cell face) between them. By using the average, the central differencing scheme aims to balance the contribution of the property values from both sides of the interface, providing a symmetric and second-order accurate representation of the diffusive flux.

This method is particularly useful in the discretization of the diffusion term in the convection-diffusion equation, where it helps in forming the algebraic equations that are used to solve for the distribution of the transported property across the domain. The central differencing scheme provides a straightforward and effective means of capturing the effects of diffusion, especially when the grid spacing is uniform, ensuring that the diffusive flux is appropriately modeled across the computational domain.

The image is based on the An Introduction to Computational Fluid Dynamics THE FINITE VOLUME METHOD Second Edition by H K Versteeg and W Malalasekera