具有最优地转平衡的混合有限元对族
A family of mixed finite element pairs with optimal geostrophic balance
AI总结：
本文提出一族用于测地网格和自适应网格细化的混合有限元对，证明其在线性旋转浅水方程中地转平衡状态精确稳态，且满足inf-sup条件，无虚假压力模，适用于数值天气预报和海洋建模。
摘要：
我们介绍了一族混合有限元对，用于测地网格和自适应网格细化，应用于数值天气预报和海洋建模。我们证明，当这些有限元对应用于线性旋转浅水方程时，地转平衡状态是精确稳态的，这意味着数值格式不会引入任何虚假的惯性重力波；这使得这些有限元对在某种意义上对于数值天气预报和海洋建模应用是最优的。我们进一步证明，这些有限元对满足inf-sup条件，这意味着它们没有虚假压力模，这些压力模会在大规模地球物理应用所需的时间尺度上污染数值解。然后，我们讨论了扩展到带旋转的不可压缩欧拉-布西内斯克方程，并表明对于线性化方程，平衡状态在任意非结构化网格上再次精确稳态。我们还表明，由这些离散化得到的离散压力泊松方程满足最优模板性质。所有这些性质使得该族离散化成为在需要非结构化网格时进行数值天气预报和大规模海洋建模应用的优秀候选者。
英文摘要：
We introduce a family of mixed finite element pairs for use on geodesic grids and with adaptive mesh refinement for numerical weather prediction and ocean modelling. We prove that when these finite element pairs are applied to the linear rotating shallow water equations, the geostrophically balanced states are exactly steady, which means that the numerical schemes do not introduce any spurious inertia-gravity waves; this makes these finite element pairs in some sense optimal for numerical weather prediction and ocean modelling applications. We further prove that these finite element pairs satisfy an inf-sup condition which means that they are free of spurious pressure modes which would pollute the numerical solution over the timescales required for large-scale geophysical applications. We then discuss the extension to incompressible Euler-Boussinesq equations with rotation, and show that for the linearised equations the balanced states are again exactly steady on arbitrary unstructured meshes. We also show that the discrete pressure Poisson equation resulting from these discretisations satisfies an optimal stencil property. All these properties make the discretisations in this family excellent candidates for numerical weather prediction and large-scale ocean modelling applications when unstructured grids are required.
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