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

Abstract

n 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 x 3 block system into a sequence of two 2 x 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 heatedmore » convection, and 3D laser-induced melt pool flow. Here, 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).« less

Authors:
 [1]; ORCiD logo [1];  [2];  [1]
+ Show Author Affiliations
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Univ. of California, Davis, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1577941
Alternate Identifier(s):
OSTI ID: 1701812
Report Number(s):
LLNL-JRNL-745515
Journal ID: ISSN 0021-9991; 900481; TRN: US2102668
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 397; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Block preconditioning; Physics-based preconditioning; Fully implicit; Newton Krylov; All speed fluid dynamics; Reconstructed discontinuous Galerkin method

Citation Formats

Weston, Brian, Nourgaliev, Robert, Delplanque, Jean -Pierre, and Barker, Andrew T. Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics. United States: N. p., 2019. Web. doi:10.1016/j.jcp.2019.07.045.
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Weston, Brian, Nourgaliev, Robert, Delplanque, Jean -Pierre, & Barker, Andrew T. Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics. United States. https://doi.org/10.1016/j.jcp.2019.07.045
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Weston, Brian, Nourgaliev, Robert, Delplanque, Jean -Pierre, and Barker, Andrew T. Thu . "Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics". United States. https://doi.org/10.1016/j.jcp.2019.07.045. https://www.osti.gov/servlets/purl/1577941.
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@article{osti_1577941,
title = {Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics},
author = {Weston, Brian and Nourgaliev, Robert and Delplanque, Jean -Pierre and Barker, Andrew T.},
abstractNote = {n 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 x 3 block system into a sequence of two 2 x 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. Here, 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).},
doi = {10.1016/j.jcp.2019.07.045},
journal = {Journal of Computational Physics},
number = C,
volume = 397,
place = {United States},
year = {2019},
month = {7}
}
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Figure 1 Figure 1: Thermal model, u(T), for equilibrium phase change.
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