标题： 无雅可比多网格预条件器应用于数值天气预报的间断伽辽金方法 显示英文标题

标题： Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods applied to Numerical Weather Prediction

摘要： 间断伽辽金（DG）方法是求解非定常可压缩流的有前景的高阶离散化方法。 在这里，我们将重点放在数值天气预报（NWP）上。 这些流动的特点是在$z$-方向上的精细分辨率和低马赫数，这使得系统变得刚性。 因此，需要隐式时间积分，并且为此需要一个快速、高度并行、低内存占用的迭代求解器来处理由此产生的代数系统。 作为基本框架，我们使用了带预条件器的不精确雅可比自由牛顿-GMRES方法。 对于低阶有限体积离散化，多重网格方法已成功应用于稳态和非稳态流体流动。 然而，对于高阶DG方法，目前缺乏此类求解器。 这促使我们研究构造适用于高阶DG离散化的雅可比自由预条件器。 该预条件器基于为DG网格子网格上定义的低阶有限体积离散化构建的多重网格方法。 我们设计了一个计算高效且质量守恒的网格间映射。 作为平滑器，使用显式龙格-库塔伪时间迭代，这些迭代可以以雅可比自由、低内存占用的方式并行实现。 我们考虑了求解二维欧拉方程和粘性流动方程的DG方法，两者都带有重力，在一种平衡良好的公式中。 在大气流动问题的DUNE-FEM软件框架中的数值实验展示了这种方法的优势。 显示英文摘要

摘要： Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in $z$-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. %The lack of efficient solvers suitable for contemporary computer architectures inhibits wider adoption of DG methods. This motivates our research to construct a Jacobian-free precondtioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.