Higher-Oder Splitting Schemes for Fluids with Variable Viscosity

2025-06-17

AI Summary

High-accuracy, robust, and efficient simulation of variable-viscosity incompressible Navier–Stokes equations remains challenging due to strong viscosity contrasts and stiffness. Method: We propose a matrix-free high-order discontinuous Galerkin (DG) spatial discretization coupled with a projection-based dual-splitting time-integration framework. This is the first extension of dual-splitting methods to non-constant viscosity regimes. We develop an hp-adaptive DG formulation tailored for strong viscosity contrasts and design a linearization strategy exploiting the saddle-point structure, accelerated by an hp-multigrid nonlinear solver. Results: Validation on benchmark problems—including backward-facing step flow—demonstrates spatiotemporal high-order accuracy, exceptional robustness, and significantly improved computational efficiency. Crucially, the fully implicit variant exhibits superior temporal stability compared to both linearized and conventional splitting schemes. The framework establishes a reliable numerical paradigm for rheologically complex flows with high viscosity contrast.

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Abstract

This article investigates matrix-free higher-order discontinuous Galerkin (DG) 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. 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) the extension of the dual splitting method originally proposed by G.E. Karniadakis et al. (J. Comput. Phys. 97, 414-443, 1991) towards non-constant viscosity, ii) a higher-order DG method for incompressible flows with variable viscosity, iii) accelerated nonlinear solver variants and suitable linearizations adopting a matrix-free $hp$-multigrid solver, and iv) 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.

Problem

• Develop higher-order DG methods for incompressible flows with variable viscosity
• Compare monolithic and splitting schemes for computational efficiency
• Extend dual splitting method to handle non-constant viscosity cases

Innovation

• Higher-order DG for variable viscosity flows
• Matrix-free hp-multigrid solver acceleration
• Dual splitting method extension for non-constant viscosity