A high-order residual-based viscosity finite element method for incompressible variable density flow

Abstract

In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; (ii) a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios.