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Nonlinear p-Multigrid Preconditioner for Implicit Time Integration of Compressible Navier–Stokes Equations with p-Adaptive Flux Reconstruction

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Abstract

Within the framework of p-adaptive flux reconstruction, we aim to construct efficient polynomial multigrid (pMG) preconditioners for implicit time integration of the Navier–Stokes equations using Jacobian-free Newton–Krylov (JFNK) methods. We hypothesise that in pseudo transient continuation (PTC), as the residual drops, the frequency of error modes that dictates the convergence rate gets higher and higher. We apply nonlinear pMG solvers to stiff steady problems at low Mach number (\(\textrm{Ma}=10^{-3}\)) to verify our hypothesis. It is demonstrated that once the residual drops by a few orders of magnitude, improved smoothing on intermediate p-sublevels will not only maintain the stability of pMG at large time steps but also improve the convergence rate. For the unsteady Navier–Stokes equations, we elaborate how to construct nonlinear preconditioners using pseudo transient continuation for the matrix-free generalized minimal residual (GMRES) method used in explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) methods, and linearly implicit Rosenbrock–Wanner (ROW) methods. Given that at each time step the initial guess in the nonlinear solver is not distant from the converged solution, we recommend a two-level \(p\{p_0\text {-}p_0/2\} \) or even \( p\{p_0\text {-}(p_0-1)\} \) p-hierarchy for optimal efficiency with a matrix-based smoother on the coarser level based on our hypothesis. It is demonstrated that insufficient smoothing on intermediate p-sublevels will deteriorate the performance of pMG preconditioner greatly. The nonlinear pMG preconditioner in this framework is found to be effective in reducing computational cost, as well as reducing the dimension of Krylov subspace for stiff systems arising from high-aspect-ratio elements and low Mach numbers. Specifically, the JFNK-pMG technique is demonstrated to be more than 5 times faster than pMG nonlinear solvers for unsteady problems. Compared to the EJ preconditioner, the pMG preconditioner can make ESDIRK and ROW methods up to 2 times faster for low-Mach-number flow and up to 1.5 times faster for highly anisotropic meshes. Moreover, the pMG preconditioner can reduce the dimension of Krylov subspace by one order of magnitude. With a pMG preconditioner, ROW methods are consistently more efficient than ESDIRK methods.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The first author would like thank Tarik Dzanic for proofreading this manuscript.

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The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Department of Aerospace Engineering, Texas A &M University, College Station, USA

    Lai Wang & Antony Jameson

  2. Department of Aeronautics, Imperial College London, London, UK

    Will Trojak

  3. Department of Ocean Engineering, Texas A &M University, College Station, USA

    Freddie Witherden

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  1. Lai Wang
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Wang, L., Trojak, W., Witherden, F. et al. Nonlinear p-Multigrid Preconditioner for Implicit Time Integration of Compressible Navier–Stokes Equations with p-Adaptive Flux Reconstruction. J Sci Comput 93, 81 (2022). https://doi.org/10.1007/s10915-022-02037-w

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