A New Finite-Difference Diffusion Scheme

A new second-order accurate, explicit diffusion scheme is presented and discussed. The scheme is derived as a weighted average of the conventional, forward-in-time, explicit diffusion scheme over one grid length and the same scheme, but over two grid lengths. Varying the weighting factors produces a family of schemes. For optimum use, a new scheme with the weighting factor dependent on the viscous stability number is proposed. It is slightly more computationally expensive than the conventional explicit scheme (typically by 25%) but is numerically stable at viscous stability numbers four times as large. Further, it is about 20% computationally less expensive than the fully implicit scheme even in the simplest one-dimensional model. This “three-level, locally implicit” scheme has been implemented in both a simple one-dimensional diffusion model and also in a complex three-dimensional large-eddy simulation model. It has been found to behave well and is profitable in both models.