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.
