论文标题
非线性P-Multigrid预处理用于可压缩Navier的隐式时间整合 - Stokes方程
Nonlinear p-multigrid preconditioner for implicit time integration of compressible Navier--Stokes equations
论文作者
论文摘要
在$ p $ - 适应通量重建的框架内，我们旨在构建有效的多项式多机（$ p $ mg）预处理器，以使用jacobian free newton-krylov（JFNK）方法使用Navier-Stokes方程的隐式时间集成。我们假设在伪瞬态延续（PTC）中，随着残差下降，决定收敛速率的误差模式的频率越来越高。我们将非线性$ p $ mg求解器应用于低马赫数（$ \ mathrm {ma} = 10^{ - 3} $）以验证我们的假设。证明，一旦残差下降了几个数量级，改善了中级$ p $ usublevels的平滑，不仅可以在大时段保持$ p $毫克的稳定性，而且还可以提高收敛速度。 For the unsteady Navier--Stokes equations, we elaborate how to construct nonlinear preconditioners using pseudo transient continuation for the matrix-free generalized minimal residual (GMRES) method used in explicit first stage, singly diagonally implicit Runge--Kutta (ESDIRK) methods, and linearly implicit Rosenbrock--Wanner (ROW) methods.鉴于在每个时间步骤中，非线性求解器中的初始猜测与收敛的解决方案不远，我们建议使用两级$ p \ {p_0 \ { - } p_0/2 \} $，甚至$ p \ {p_0 \ {p_0 \ {p_0 \ { - } { - }（ - }（p_0-1）（p_0-1）（p_0-1）\} $ $ p $ - $ $ - $ - $ - $ - $ -HERARTY，基于我们的假设。已经证明，在中间$ p $ usublevels上的平滑不足将大大恶化$ p $毫克预处理的性能。 （请参阅论文中的完整摘要。）
Within the framework of $ p $-adaptive flux reconstruction, we aim to construct efficient polynomial multigrid ($p$MG) 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 $p$MG solvers to stiff steady problems at low Mach number ($\mathrm{Ma}=10^{-3}$) 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 $ p $MG 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 $p$MG preconditioner greatly. (See full abstract in the paper.)
