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

Abstract

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.

Figures (7)

• Figure 1: Initial vertical profiles for the 1D test case, with a potential temperature perturbation overlaying a state of hydrostatic balance.
• Figure 2: Energy conservation error (left), Helmholtz operator condition number (center) and number of nonlinear iterations to convergence (right) for the three different preconditioners using exact matrix inverses.
• Figure 3: Energy conservation error (left), Helmholtz operator condition number (center) and residual error at the final Newton iteration (right) for the three different preconditioners using approximate matrix inverses and four Newton iterations per time step.
• Figure 4: Potential temperature perturbation from the mean initial state, $\theta_p = \theta-\theta_m(t=0)$ at time $3000s$ for the 2D non-hydrostatic gravity wave test case for the new (top) and Charney-Phillips (bottom) formulations. For the new formulation the potential temperature perturbation is projected from discontinuous space $\mathbb{W}_3$ to continuous space $\mathbb{W}_0$. Colors range from $-0.00146^{\circ}K$ to $+0.00252^{\circ}K$. Vertical axes is scaled by a factor of 10 with respect to the horizontal.
• Figure 5: Energy conservation error (top), and residual errors after four nonlinear iterations (bottom) for the 2D non-hydrostatic gravity wave test case for the new formulation and the material transport of $\theta$ on the Charney-Phillips grid.