We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches.

翻译：我们提出了一种用于垂直切片可压缩欧拉方程的兼容有限元离散格式，其阶次为次最低阶（即压力空间采用双线性不连续函数）。该方程在时间上采用全隐式时间步进方案进行数值积分，并通过以线条平滑器预处理的整体式GMRES求解器进行求解。所述线条平滑器仅涉及局部运算，因此适用于并行域分解。该方法允许采用任意大的时间步长，但在大库朗数极限下，其迭代次数与库朗数呈线性关系。该求解器方法基于Firedrake实现，并利用了PETSc的加法施瓦茨预处理框架。我们通过一组可与其它方法进行比较的标准测试案例，验证了该方案的鲁棒性。