Collaborative Research: Efficient Unstructured Discontinuous Galerkin Methods for Global Nonhydrostatic Atmospheric Modeling

本文提出了一种有效的非结构不连续伽辽金（DG）方法，用于全球非流体静力大气模拟。随着数值天气预报模式水平分辨率的提高，有利于模拟非流体静力运动。深层对流云的动力学是非流体静力的，这些云对更大尺度的气流产生重要的反馈，而这些反馈很难参数化。为了表示这样的云和类似的小尺度过程，全球大气模式解出的方程必须从流体静力公式过渡到非流体静力公式。当一个人进入非流体静力状态时，会出现一系列具有挑战性的、重要的数值问题，包括：1)开发有效的时间积分器和/或“隔音”方程集，以应对由控制方程支持的快速声波；II)有效地解决多尺度流动特征，这可能需要自适应网格细化（AMR）；III)构建保留所有重要量并可用于满足上述两个条件的空间离散化方法。第(I)项将通过为我们的控制方程和离散空间算子设计一套隐式-显式（IMEX）时间积分器来解决，以允许更大的显式时间步长和更好的条件隐式部分。此外，在气象学相关的测试案例中，将完全可压缩（欧拉）方程与隔音系统（非弹性、伪不可压缩）进行比较。第（II）项将通过测试两种形式的AMR来解决：符合和不符合方法。一致性方法通常与连续离散化方法一起使用，由于AMR本质上是一种独立的方法，因此不需要对偏微分方程（PDE）求解器进行修改。不符合方法传统上用于不连续方法，但也可以用于连续方法。不符合方法以无缝的方式将PDE求解器与AMR方法耦合在一起，因此需要修改PDE求解器。项目（III）将通过DG方法来解决，该方法能够提供所有预测变量的局部和全局守恒，并且也非常适合处理可能由潮湿变量引入的强梯度（例如，不连续面）。长期天气预报和地球气候研究使用计算机模拟整个地球的天气。单个的雷暴和类似的深云团产生降雨，并在整个大气中垂直输送水分、热量和动量。由于计算机能力的限制，全球模拟无法使用描述流体运动的基本方程来计算这些局部云的行为。相反，这些云对全球大气的净影响是用必要的粗略近似值“参数化”的。计算机体系结构的最新进展最终使在全球模型中模拟大型云成为可能，但仍有一些必须克服的重大挑战。全球模式所依据的数学方程必须更接近流体运动的基本方程，必须确定一种有效的灵活结构来保存描述全球大气状态的数据。这项研究解决了这两个挑战。

This is a proposal to develop efficient unstructured discontinuous Galerkin (DG) methods for global non-hydrostatic atmospheric modeling. As the horizontal resolution of numerical weather prediction models increases, it becomes advantageous to model non-hydrostatic motions. The dynamics of deep convective clouds are non-hydrostatic and these clouds produce important feedbacks on the larger-scale flow that are difficult to parameterize. To represent such clouds and similar small-scale processes, the equations solved by global atmospheric models must transition from hydrostatic to non-hydrostatic formulations. A host of challenging, non-trivial numerical problems arise when one enters the non-hydrostatic regime, including: I) developing efficient time-integrators and/or "soundproof" equation sets to confront the fast acoustic waves supported by the governing equations; II) effectively resolving multi-scale flow features, that may require adaptive mesh refinement (AMR); and III) constructing spatial discretization methods that conserve all important quantities and can be exploited to satisfy the above two conditions. Item (I) will be addressed by designing a suite of implicit-explicit (IMEX) time-integrators for our governing equations and discrete spatial operators to allow larger explicit time-steps with better conditioned implicit parts. In addition, the fully compressible (Euler) equations will be compared to sound-proof systems (anelastic, pseudo-incompressible) in meteorologically relevant test cases. Item (II) will be addressed by testing two forms of AMR: conforming and nonconforming methods. Conforming methods are traditionally used with continuous discretization methods and require no modifications to the partial differential equation (PDE) solver since the AMR is essentially a separate method. Nonconforming methods are traditionally used with discontinuous methods but can also be used with continuous methods. Nonconforming methods couple the PDE solver with the AMR approach in a seamless fashion thereby requiring modification of the PDE solver. Item (III) will be addressed by virtue of the DG method that is capable of delivering both local and global conservation of all prognostic variables and is also well suited for handling strong gradients (e.g., discontinuities) that may be introduced by the moist variables. Long-range weather forecasting and studies of the Earth's climate use computers to simulate the weather over the entire globe. Individual thunderstorms and clusters of similar deep clouds produce rain and transport moisture, heat, and momentum vertically throughout the atmosphere. Limitations in computer power have prevented global simulations from using the fundamental equations describing fluid motion to calculate the behavior of these localized clouds. Instead, the net effects of such clouds on the global atmosphere are "parameterized" using necessarily crude approximations. The latest advancements in computer architecture have finally made it possible to simulate large clouds in global models, but there are major challenges that must be overcome. The mathematical equations on which global models are based must become better approximations to the fundamental equations of fluid motion, and an efficient flexible structure must be determined to hold the data describing the state of the global atmosphere. This research addresses both of these challenges.