Center for Turbulence Research Proceedings of the Summer Program 2008 97 A comparison of various equal-order interpolation methodologies using the method of manufactured solutions By S. P. Domino† This paper represents a first step to providing a comparison between a suite of ap- proximate projection methods and stabilized equal-order, i.e., the primitive variables are collocated with an equal-order interpolation basis, fully coupled methods. The paper first reviews splitting and stabilization errors that appear in many common approximate pro- jection methods. The approximate pressure projection analysis demonstates a conflicted role for the pressure stabilization scaling parameter. This parameter must be chosen to control both splitting errors and stabilization errors. However, for fully coupled schemes splitting errors are not of consequence. Therefore, a stabilization parameter can be chosen to most appropriately control stabilization errors. A fully coupled equal-order scheme is proposed with two forms of the stabilization parameter. A set of manufactured solutions are run to verify the accuracy of this new method. 1. Introduction In the previous Center for Turbulence Research Summer Program, the role of explicit stabilization in the context of common equal-order pressure projection schemes was out- lined (Domino 2006). A family of projection methods was defined based on various time scale choices. Three common approximate projection methods were presented in detail. A formal manufactured solution was presented to verify that the standard time step stabi- lization parameter leads to first-order time accuracy despite the underlying second-order time integration scheme. To review, the analysis of a given computational fluids algorithm begins with the discrete momentum and continuity equations written in matrix form. The matrix A contains discrete, linearized contributions to the momentum equations from the time derivative, convection and diffusion terms,  A G D 0  u n+1 p n+1/2  =  f b  . (1.1) The discrete nodal gradient and nodal divergence are G and D, respectively. The function f contains the additional terms for the momentum equations, e.g., body force terms, while the function b can contain the appropriate terms for a non-solenoidal velocity field, i.e., −  ∂ρ ∂t dV . Finally, variable density aspects can be provided in the exact form of D. As previously described in Domino (2006) the full set of splitting and stabilization errors in the context of a fourth-order stabilized general approximate projection method are given by: † Computational Thermal and Fluid Mechanics, Sandia National Laboratories