Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggering
CoRR(2024)
摘要
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.
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