Towards High-Order Fluctuation-Splitting Schemes for Navier-Stokes Equations

This paper reports progress towards high-order fluctuation-splitting schemes for the Navier-Stokes Equations. High-order schemes we examined previously are all based on gradient reconstruction, which may result in undesired mesh-dependency problem due to the somewhat ambiguous gradient reconstruction procedures. Here, we consider schemes for P2 elements in order to eliminate the need for such gradient reconstruction. For pure advection, a P2 version of the LDA scheme is derived from a constrained least-squares minimization. This scheme is fourth-order accurate. For pure diffusion, a P2 Galerkin scheme is derived from a minimization principle, which turns out to be equivalent to applying Richardson’s extrapolation technique to the standard second-order Galerkin scheme. This scheme is again fourth-order accurate. Finally, strategies for integrating P2 advection and diffusion schemes to develop uniformly accurate P2 schemes are discussed.