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的（近似）雅可比矩阵分解为关于气压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，该方程采用洛伦兹网格上以通量形式表示的位温，并使用混合有限元方法。该离散格式能够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，所得预处理器与现有处理Charney-Phillips网格上位温物质形式传输的预处理器具有相似的块结构，并且对于一维热泡算例，其效率和稳定性均优于该预处理器及先前基于洛伦兹网格通量形式密度加权位温传输的亥姆霍兹预处理器。该新预处理器进一步通过了垂直切片几何构型下的标准二维测试算例验证。