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

本文提出了一种用于全球非静力大气模拟的高效非结构间断伽辽金(DG)方法。随着数值天气预报模式水平分辨率的提高，对非静力运动的模拟变得更加有利。深对流云的动力学是非静力的，这些云对更大尺度的流动产生重要的反馈，这是很难参数化的。为了表示这种云和类似的小尺度过程，全球大气模型求解的方程必须从流体静力公式过渡到非流体静力公式。当人们进入非静力状态时，出现了许多具有挑战性的、非平凡的数值问题，包括：i)开发有效的时间积分器和/或“隔音”方程组来对抗由控制方程支持的快速声波；ii)有效地解析多尺度流动特征，这可能需要自适应网格加密(AMR)；以及iii)构造保存所有重要量并可用于满足上述两个条件的空间离散方法。项目(I)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，以允许具有更好条件的隐式部分的更大的显式时间步长。此外，完全可压缩(Euler)方程将在气象相关测试案例中与隔音系统(滞弹性、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。协调方法传统上与连续离散方法一起使用，不需要对偏微分方程(PDE)求解器进行修改，因为AMR本质上是一种独立的方法。非一致性方法传统上与不连续方法一起使用，但也可以与连续方法一起使用。非协调方法以无缝方式将PDE求解器与AMR方法耦合，从而需要修改PDE求解器。项目(Iii)将通过DG方法解决，该方法能够提供所有预测变量的局部和全局守恒，并且也非常适合于处理可能由潮湿变量引入的强梯度(例如，不连续性)。长期天气预报和对地球气候的研究使用计算机来模拟整个地球的天气。单独的雷暴和类似的深层云团会产生降雨，并在大气中垂直输送水分、热量和动量。计算机能力的限制使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。取而代之的是，这些云对全球大气的净影响是使用必然是粗略的近似来“参数化”的。计算机体系结构的最新进展终于使在全球模型中模拟大型云成为可能，但仍有一些重大挑战必须克服。全球模型所基于的数学方程必须更好地近似于流体运动的基本方程，并且必须确定一个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.