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Summary Report No. 84


Well Balanced Finite Volume Methods for Nearly Hydrostatic Flows
N. Botta, R. Klein, S. Langenberg, S. Lützenkirchen (August 2003)

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.


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