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Abstract

Recent trends towards the construction of mass and energy conserving, non-hydrostatic, and fully compressible flow models for purposes of numerical weather prediction and regional climate modelling motivate the present work. In this context, a proper numerical representation of the dominant hydrostatic balance is of crucial importance: unbalanced truncation errors can induce unacceptable spurious motions, in particular near steep topography.
In this paper we develop a new strategy for the construction of discretizations that are "wellbalanced" with respect to dominant hydrostatics. The popular subtraction of a "hydrostatic background state" is avoided by the introduction of local, time dependent hydrostatic reconstructions. Balanced discretizations of the pressure gradient and the gravitation source term are achieved through a judicious implementation of a "discrete Archimedes’ buoyancy principle".
This strategy is applied to extend an explicit standard finite volume Godunov-type scheme for compressible flows with minimal modifications. We plan to address a large time step semi-implicit version of the scheme in future work. The resulting method inherits its conservation properties from the underlying base scheme and has three distinct and desirable features: (i) It is exactly balanced, even on curvilinear grids, for a large class of near-hydrostatic flows. (ii) It directly solves the full compressible flow equations while avoiding the non-local, possibly time-consuming computation of a (slowly time-dependent) background state. (iii) It is robust against details of the implementation, such as the choice of slope limiting functions, or the particulars of boundary condition discretizations.