Methods for upwinding the potential vorticity in a compatible finite element discretisation of the rotating shallow water equations are studied. These include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a recent approach where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent, resulting in a turbulent state where the potential enstrophy is more strongly damped than for the other schemes. While the SUPG scheme is widely known to be consistent, since it modifies the test functions only, the downwinded trial function formulation results in the advection of downwind corrections. Results of the SUPG and downwinded trial function schemes are very similar in terms of both potential enstrophy conservation and turbulent spectra. The main difference between these schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme. We also present formulations by which potential enstrophy is exactly integrated at each time level. Results using these formulations are observed to be stable in the absence of dissipation, despite the uncontrolled aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient $\mathcal{O}(100)$ times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.

翻译：研究了旋转浅水方程相容有限元离散中势涡度迎风格式的方法。这些方法包括著名的预期势涡度法（APVM）、流向迎风Petrov-Galerkin（SUPG）法，以及一种在参考单元内下游评估试验函数的新近方法。所有迎风格式均守恒势涡度和能量，因为方程的反对称结构得以保留。APVM对势涡度产生一个对称正定的耗散且非一致的修正，导致湍流状态中势涡度比其他格式受到更强的阻尼。虽然SUPG格式因仅修改检验函数而被广泛认为是一致的，但下游试验函数格式会导致下游修正的平流。SUPG与下游试验函数格式在势涡度守恒和湍流谱方面的结果非常相似。这些格式的主要区别在于能量守恒和残差误差。若仅进行两次非线性迭代，则下游试验函数格式的能量守恒误差有所改善，反映出其残差误差小于SUPG格式。我们还提出了在每个时间层精确积分势涡度的公式。使用这些公式的结果显示，尽管存在网格尺度湍流的无控混叠，但在无耗散情况下仍保持稳定。采用该公式以及系数比常规值小$\mathcal{O}(100)$倍的APVM，得到的湍流谱在网格尺度上优于存在不稳定势涡度误差的SUPG和下游试验函数格式。