比较地球物理流流体的变向顺流计划及其在潜在富营养分化中的应用 (A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations)

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：正在研究在可兼容的有限元素下调可变浅水方程式中潜在变色体的方法。这些方法包括众所周知的预期潜在园艺方法(APVM)、流向上风Petrov-Galerkin(SUPG)方法,以及最近在参考元素下游对试验功能进行评估的方法。在所有情况中, 上风方案保留了潜在变色体和能量, 因为等方的反对称结构得到了保存。 APVM 导致对潜在变异性且前后不一的累进性进行对称明确校正, 导致动荡状态, 潜在变异性比其他方案更强。虽然SUPG计划由于仅修改测试功能, 测试功能在下游期间对潜在变色体和能量。 SUPG和下调试功能的结果在潜在变色体中非常相似( 潜在变色保存和变形变色变色变色) 。这些变变变变变变变变变的变形的主要差异在于时间, 其变色变变变的变变变的变的变变变变变的变的变的变的变的变的变的变的变变变的变, 的变变变变变的变的变的变的变变的变的变的变的变的变的变的变的变的变的变的变变变变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变, 。