Abstract The simulation of the transport problem on the sphere is crucial in the numerical modelling of the transport of trace constituents in atmospheric models. One major issue in the numerical simulation is the ability of the solver to obtain accurate constraint-preserving solutions, i.e., ensuring the predicted solution to stay within the physical range. In this paper, we develop and study a variational inequality (VI) based optimization methodology for constructing a new transport model that naturally satisfies this restriction by removing over- and under-shoots of the solution. With the use of the cubed-sphere mesh, we present an explicit-first-step, single-diagonal coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with an adaptive time step control strategy for the fully implicit temporal integration of the variational inequality problem. And then the resulting nonlinear system arising at each time step is solved by using some nonlinear and linear fast solver technologies, including a variant of inexact Newton methods, i.e., the active-set reduced-space (ASRS) method, and the domain decomposition based preconditioners with a novel analytical Jacobian matrix. A set of tracer transport test cases based on the variational inequality model is presented to demonstrate the efficiency and robustness of the proposed method.

中文翻译：

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基于活动集缩减空间算法的球体变分不等式输运模型

摘要 球体输运问题的模拟对于大气模型中痕量成分输运的数值模拟至关重要。数值模拟中的一个主要问题是求解器获得精确约束保持解的能力，即确保预测解保持在物理范围内。在本文中，我们开发并研究了一种基于变分不等式 (VI) 的优化方法，用于构建一个新的传输模型，该模型通过消除解决方案的过冲和下冲来自然地满足这一限制。通过使用立方球体网格，我们提出了一个明确的第一步，单对角线系数，具有自适应时间步长控制策略的对角隐式 Runge-Kutta (ESDIRK) 方法用于变分不等式问题的完全隐式时间积分。然后通过使用一些非线性和线性快速求解器技术来求解在每个时间步产生的非线性系统，包括不精确牛顿方法的变体，即活动集缩减空间（ASRS）方法，以及基于域分解的具有新颖的分析雅可比矩阵的预处理器。提出了一组基于变分不等式模型的示踪剂传输测试用例，以证明所提出方法的效率和鲁棒性。活动集缩减空间 (ASRS) 方法，以及基于域分解的预处理器，具有新颖的分析雅可比矩阵。提出了一组基于变分不等式模型的示踪剂传输测试用例，以证明所提出方法的效率和鲁棒性。活动集缩减空间 (ASRS) 方法，以及基于域分解的预处理器，具有新颖的分析雅可比矩阵。提出了一组基于变分不等式模型的示踪剂传输测试用例，以证明所提出方法的效率和鲁棒性。