An improved two-time-level split-explicit integration scheme for non-hydrostatic compressible models

SummaryA new two-time-level split-explicit time integration scheme for the use in non-hydrostatic compressible modelling is presented. It is demonstrated that the scheme is numerically stable and has a smaller splitting error than other comparable split-explicit schemes. This error is due to the combination of advection and fast-wave terms in the numerical scheme.To outline where the splitting error occurs and how it acts within the splitting mechanism, a short review of existing split-explicit time integration methods is given. An in-depth analysis of the eigenvalues of several two-time-level schemes is performed showing that instabilities are associated with the splitting error term. The term has different signs for forward moving and backward moving waves, causing exponential growing or decaying. This unwanted characteristic is not prevented by a diffusive term in general, but might be counteracted by schemes using an estimate of the fast waves at the midpoint of the time increment.The importance of this fast-waves midpoint estimate leads to the formulation of a class of split-explicit two-time-level schemes. Within this framework any forward-in-time and stable advection scheme might be combined with the fast-waves terms in the splitting algorithm.The new method is implemented in the non-hydrostatic model LM of DWD. Some test cases are presented, indicating that the new scheme has the potential to be used in an operational environment.

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