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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).
##### MSC:
 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
##### Software:
AUSM; BoomerAMG; hypre; METIS; ParMETIS; PETSc
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