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.

翻译：大气模型的隐式求解器通常通过求解预处理系统来加速。对于块预处理器，这通常涉及将耦合系统线性化产生的（近似）雅可比矩阵分解为某个压力函数的亥姆霍兹方程。本文提出了一种针对可压缩欧拉方程的预处理器，该方程在洛伦兹网格上采用通量形式表示位温，并使用混合有限元。此公式允许空间离散化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，所得预处理器被证明具有与现有预处理器相似的块结构，该现有预处理器用于查尼-菲利普斯网格上位温的物质形式输运。进一步证明，对于一维热泡配置，这种新公式比查尼-菲利普斯网格上的位温物质形式输运，以及先前洛伦兹网格上密度加权位温通量形式输运的亥姆霍兹预处理器，都更高效且更稳定。该新预处理器还在垂直切片几何中的标准二维测试案例中得到了进一步验证。