Higher-Order Discontinuous Galerkin Splitting Schemes for Fluids with Variable Viscosity

TL;DR

This paper develops and compares higher-order discontinuous Galerkin schemes for simulating incompressible flows with variable viscosity, focusing on computational efficiency, stability, and solver performance in complex fluid scenarios.

Contribution

Introduces a novel higher-order DG discretization for variable viscosity flows and compares monolithic and splitting solvers using a matrix-free multigrid approach.

Findings

Splitting schemes are computationally efficient for certain problems.

Implicit schemes offer better stability at higher computational costs.

The proposed methods accurately handle viscosity contrasts in numerical tests.

Abstract

This article investigates matrix-free higher-order discontinuous Galerkin discretizations of the Navier--Stokes equations for incompressible flows with variable viscosity. The viscosity field may be prescribed analytically or governed by a rheological law, as often found in biomedical or industrial applications. The DG discretization of the adapted second-order viscous terms is carried out via the symmetric interior penalty Galerkin method, obviating auxiliary variables. Based on this spatial discretization, we compare several linearized variants of saddle point block systems and projection-based splitting time integration schemes in terms of their computational performance. Compared to the velocity-pressure block-system for the former, the splitting scheme allows solving a sequence of simple problems such as mass, convection-diffusion and Poisson equations. We investigate under which conditions the improved temporal stability of fully implicit schemes and resulting expensive nonlinear solves outperform the splitting schemes and linearized variants that are stable under hyperbolic time step restrictions. The key aspects of this work are i) a higher-order DG discretization for incompressible flows with variable viscosity, ii) accelerated nonlinear solver variants and suitable linearizations adopting a matrix-free $hp$-multigrid solver, and iii) a detailed comparison of the monolithic and projection-based solvers in terms of their (non-)linear solver performance. The presented schemes are evaluated in a series of numerical examples verifying their spatial and temporal accuracy, and the preconditioner performance under increasing viscosity contrasts, while their efficiency is showcased in the backward-facing step benchmark.