AI 中文总结
本文提出AdHImEx，一种新的自适应隐式-显式输运时间步进方案，在大Courant数下实现二阶精度且每步仅需一次矩阵求解，并引入阶段依赖散度处理消除虚假散度，显著提升大气输运的精度与效率。
AI 中文摘要
自适应隐式-显式（AdImEx）时间步进为大时间步长的质量守恒输运提供了稳定性，使其在天气和气候预测中平流数值表示方面具有吸引力。现有的AdImEx输运方案在大时间步长、隐式区域中仅具有一阶精度，并且每个时间步长可能需要多次稀疏矩阵求解。本工作引入了AdHImEx，一种同时解决这两个局限性的新AdImEx方案。AdHImEx在大Courant数下提供二阶精度，同时每个时间步长仅需一次矩阵求解。它是一种Runge-Kutta方案，融合了三阶精确显式时间步进和Crank-Nicolson隐式时间步进，为Courant数高达100提供了数值验证的稳定性。AdHImEx专为以显式为主的流动设计，仅在出现大Courant数的区域激活隐式时间步进。在小Courant数区域，它保留了显式时间步进的完整效率和三阶精度。在本工作中，它与五阶精确有限体积空间离散相结合以实现质量守恒。本工作的第二个贡献是对散度算子的一种新颖的阶段依赖处理，尽管隐式和显式Runge-Kutta阶段时间步长不同且空间上隐式程度变化，仍能保持常数性，消除了虚假散度。这一创新适用于其他AdImEx时间步进方案。收敛性、输运和效率测试表明，相比之前的一阶AdImEx方案有显著改进，相位和振幅误差小，在具有大Courant数的局部细化网格上输运准确，并为大气输运提供了有前景的效率特性。
英文摘要
Adaptively Implicit-Explicit (AdImEx) time stepping provides stability for large time steps for mass-conservative transport, making it attractive for the numerical representation of advection in weather and climate prediction. Existing AdImEx transport schemes become first-order accurate in the large-time-step, implicit regime and may require multiple sparse matrix solutions per time step. This work introduces AdHImEx, a new AdImEx scheme that addresses both limitations simultaneously. AdHImEx provides second-order accuracy for large Courant numbers while requiring only a single matrix solution per time step. It is a Runge-Kutta scheme that blends third-order accurate explicit time stepping and Crank-Nicolson implicit time stepping to provide numerically verified stability for Courant numbers up to 100. AdHImEx is designed for flows that are predominantly explicit, activating implicit time stepping only where large Courant numbers occur. It retains the full efficiency and third-order accuracy of the explicit time stepping in regions with small Courant numbers. In this work, it is combined with a fifth-order accurate finite-volume discretisation in space for mass conservation. A second contribution of this work is a novel stage-dependent treatment of the divergence operator that preserves constancy despite different implicit and explicit Runge-Kutta stage time steps and spatially varying implicitness, removing spurious divergence. This innovation is applicable to other AdImEx time stepping schemes. Convergence, transport, and efficiency tests demonstrate substantial improvements over previous first-order AdImEx schemes, little phase and amplitude error, accurate transport on locally refined meshes with large Courant numbers, and promising efficiency characteristics for atmospheric transport.
