This paper provides an overview of the approximate projection methods commonly used in the unstructured finite volume community that have been implemented within SIERRA/Fuego. The paper outlines splitting and stabilization errors that appear in many common approximate projection methods. Smoothing errors, or stabilization errors, can appear in both projection and fully coupled algorithms and, as will be shown, can have drastic effects on the formal time accuracy of a chosen algorithm. A time-dependent manufactured solution is presented and used to verify formal order of accuracy for a suite of approximate projection algorithms. For a commonly used set of projection scaling time scales, first-order accuracy is demonstrated regardless of time integration scheme, i.e., backward Euler or Crank-Nicholson. A new class of approximate projection algorithms is presented that circumvents the stabilization error. This method is based on the classic pressure projection method, where the pressure is computed based on solving a Pressure Poisson Equation that is derived by taking the divergence of the momentum equation. Results show, however, that this algorithm (as implemented) is not competitive based on the significant expense of solving this second Poisson equation.

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@inproceedings{DominoTowardVO,
title={Toward verification of formal time accuracy for a family of approximate projection methods using the method of manufactured solutions},
author={By S. P. Domino}
}