Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics

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 3×3 block system into a sequence of two 2×2 block systems: velocity-pressure, vP, and velocity-temperature, vT. We compare the performance of the vP-vT 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 vP-vT 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). •A scalable and robust block preconditioner is developed for fully-implicit all-speed compressible flow with phase change.•The block preconditioner uses the Schur complement to reduce the fully coupled 3×3 Jacobian to a set of two 2×2 systems.•The developed Schur complement block preconditioner robustly converges systems arising from large CFL/Fourier numbers.•The preconditioned Newton-Krylov solver scales well both algorithmically and in parallel for all-speed compressible flow with melt convection.