Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics. (English) Zbl 1453.76084

Summary: In this paper, we introduce a multigrid block-based preconditioner for solving linear systems arising from a Discontinuous Galerkin discretization of the all-speed Navier-Stokes equations with phase change. The equations are discretized in conservative form with a reconstructed Discontinuous Galerkin (rDG) method and integrated with fully-implicit time discretization schemes. To robustly converge the numerically stiff systems, we use the Newton-Krylov framework with a primitive-variable formulation (pressure, velocity, and temperature), which is better conditioned than the conservative-variable form at low-Mach number. In the limit of large acoustic CFL number and viscous Fourier number, there is a strong coupling between the velocity-pressure system and the linear systems become non-diagonally dominant. To effectively solve these ill-conditioned systems, an approximate block factorization preconditioner is developed, which uses the Schur complement to reduce a \(3 \times 3\) block system into a sequence of two \(2 \times 2\) block systems: velocity-pressure, \(\mathbf{v} P\), and velocity-temperature, \(\mathbf{v} T\). We compare the performance of the \(\mathbf{v} P\)-\(\mathbf{v} T\) Schur complement preconditioner to classic preconditioning strategies: monolithic algebraic multigrid (AMG), element-block SOR, and primitive variable block Gauss-Seidel. The performance of the preconditioned solver is investigated in the limit of large CFL and Fourier numbers for low-Mach lid-driven cavity flow, Rayleigh-Bénard melt convection, compressible internally heated convection, and 3D laser-induced melt pool flow. Numerical results demonstrate that the \(\mathbf{v} P\)-\(\mathbf{v} T\) Schur complement preconditioned solver scales well both algorithmically and in parallel, and is robust for highly ill-conditioned systems, for all tested rDG discretization schemes (up to 4th-order).


76M10 Finite element methods applied to problems in fluid mechanics
65F08 Preconditioners for iterative methods
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76N06 Compressible Navier-Stokes equations
80A22 Stefan problems, phase changes, etc.
80A17 Thermodynamics of continua
Full Text: DOI


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