Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggering

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.