A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow

• Published in: Journal of computational physics Vol. 251; pp. 480 - 486
• Main Authors: Steinmoeller, D.T., Stastna, M., Lamb, K.G.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.10.2013
• Subjects: Boussinesq approximation; Computational fluid dynamics; Computer simulation; Discontinuous Galerkin method; Discretization; Fluid flow; Galerkin methods; High-order element methods; Incompressible flow; Incompressible Navier–Stokes equations; Non-hydrostatic flow; Operators; Projection; Projection methods; Boussinesq approximation; Discontinuous Galerkin method; Incompressible Navier–Stokes equations; High-order element methods; Projection methods; Non-hydrostatic flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2013.05.036

This note reports on the issue of spurious compressibility artifacts that can arise when the popular pressure projection (PP) method is used for unsteady simulations of incompressible flow using the symmetric interior penalty discontinuous Galerkin (SIP-DG) method. Through a spectral analysis of the projection operator’s SIP-DG discretization, we demonstrate that the eigenfunctions of the operator do not form a basis that allows for the correct enforcement of the incompressibility constraint. This short-coming can cause numerical instabilities for inviscid, advection-dominated, and density stratified flow simulations, especially for long-time integrations and/or under-resolved situations. To remedy this problem, we propose a local post-processing projection that enforces incompressibility exactly to allow for stable and robust long-time integrations.