使用带 Lorenz 交错排列的混合有限元对可压缩 Euler 方程进行 Helmholtz 预条件处理
Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggering
AI总结：
本文提出一种基于混合有限元和 Lorenz 网格的可压缩 Euler 方程 Helmholtz 预条件子，通过引入干热力学熵作为辅助变量，在保证能量和位温方差守恒的同时，实现比现有方法更高效稳定的求解。
摘要：
大气模型的隐式求解器通常通过求解预条件系统来加速。对于块预条件子，这通常涉及将耦合系统线性化得到的（近似）Jacobian 矩阵分解为某个压力函数的 Helmholtz 方程。本文针对可压缩 Euler 方程提出了一种预条件子，该方程在 Lorenz 网格上使用混合有限元，并采用位温的通量形式表示。这种公式允许空间离散同时守恒能量和位温方差。通过引入干热力学熵作为代数系统求解的辅助变量，所得的预条件子被证明具有与 Charney-Phillips 网格上位温物质形式输运的现有预条件子相似的块结构。对于一维热泡构型，这一新公式还被证明比 Charney-Phillips 网格上位温的物质形式输运以及先前用于 Lorenz 网格上密度加权位温通量形式输运的 Helmholtz 预条件子更高效且更稳定。该新预条件子进一步在垂直切片几何中的标准二维测试算例上得到了验证。
英文摘要：
Implicit solvers for atmospheric models are often accelerated via the solution of a preconditioned system. For block preconditioners this typically involves the factorisation of the (approximate) Jacobian resulting from linearization of the coupled system into a Helmholtz equation for some function of the pressure. Here we present a preconditioner for the compressible Euler equations with a flux form representation of the potential temperature on the Lorenz grid using mixed finite elements. This formulation allows for spatial discretisations that conserve both energy and potential temperature variance. By introducing the dry thermodynamic entropy as an auxiliary variable for the solution of the algebraic system, the resulting preconditioner is shown to have a similar block structure to an existing preconditioner for the material form transport of potential temperature on the Charney-Phillips grid. This new formulation is also shown to be more efficient and stable than both the material form transport of potential temperature on the Charney-Phillips grid, and a previous Helmholtz preconditioner for the flux form transport of density weighted potential temperature on the Lorenz grid for a 1D thermal bubble configuration. The new preconditioner is further verified against standard two dimensional test cases in a vertical slice geometry.
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