适用于确定性和随机旋转浅水波方程的保能相容时空有限元方法
Energy conserving compatible space--time FEM for deterministic and stochastic rotating shallow water
AI总结：
该研究提出一种适用于确定性和随机旋转浅水波方程的全离散保能时空有限元方法，采用相容空间离散与连续 Petrov-Galerkin 时间离散，结合整体块光滑子迭代求解器，经数值测试验证了其收敛、守恒与鲁棒性。
摘要：
我们提出了旋转浅水波方程的全离散保能离散格式，该格式在空间和时间上均可达到任意阶精度。采用时空有限元离散，其空间部分基于相容有限元方法，时间部分采用连续 Petrov-Galerkin 格式。为实现精确能量守恒，哈密顿导数的投影辅助变量是必要的。我们提供了一种基于整体块光滑子的迭代求解方法，该方法可使牛顿法高效收敛，从而将求解器导致的能量误差降至极小。我们将该构造扩展至基于李变换的保能随机强迫公式，随后通过数值测试验证了其收敛性、守恒性与鲁棒性。
英文摘要：
We present a fully discrete energy conserving discretisation of the rotating shallow water equations with arbitrary order in space and time. A space time finite element discretisation is used, built from compatible finite element methods in space and continuous Petrov-Galerkin schemes in time. Projected auxiliary variables for Hamiltonian derivatives are necessary for exact energy conservation. We provide an iterative solver approach based on monolithic patch smoothers, which allows us to efficiently converge a Newton method sufficiently that energy errors due to the solver are very small. We extend this construction to the energy conserving Stochastic Forcing by Lie Transport formulation. Then we demonstrate the convergence, conservation and robustness in numerical tests.
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