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 for 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, and to be more efficient and stable than either this or a previous Helmholtz preconditioner for the flux form transport of density weighted potential temperature on the Lorenz grid for a one dimensional thermal bubble configuration. The new preconditioner is further verified against standard two dimensional test cases in a vertical slice geometry.

翻译：大气模型隐式求解器通常通过预处理系统的求解来加速。对于块预处理方法，这通常涉及将耦合系统的（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方法，该方法在洛伦兹网格上采用混合有限元技术处理位温通量形式表示。该公式允许实现同时守恒能量和位温方差的空间离散化。通过引入干热力学熵作为代数系统求解的辅助变量，所得预处理器与查尼-菲利普斯网格上物质形式位温输运的已有预处理器具有相似的块结构，且在一维热泡配置下，其效率和稳定性均优于该预处理器以及洛伦兹网格上密度加权位温通量形式输运的先前亥姆霍兹预处理器。新预处理器在垂直切片几何中的标准二维测试算例中得到进一步验证。