Operator-Split Runge-Kutta-Rosenbrock Methods for Nonhydrostatic Atmospheric Models

This paper presents a new approach for discretizing the nonhydrostatic Euler equations in Cartesian geometry using an operator-split time-stepping strategy and unstaggered upwind finite-volume model formulation. Following the method of lines, a spatial discretization of the governing equations leads to a set of coupled nonlinear ordinary differential equations. In general, explicit time-stepping methods cannot be applied directly to these equations because the large aspect ratio between the horizontal and vertical grid spacing leads to a stringent restriction on the time step to maintain numerical stability. Instead, an A-stable linearly implicit Rosenbrock method for evolving the vertical components of the equations coupled to atraditionalexplicitRunge‐Kutta formulainthehorizontalisproposed.Uptothird-ordertemporalaccuracy is achieved by carefully interleaving the explicit and linearly implicit steps. The time step for the resulting Runge‐Kutta‐Rosenbrock‐type semi-implicit method is then restricted only by the grid spacing and wave speed in the horizontal. The high-order finite-volume model is tested against a series of atmospheric flow problems to verify accuracy and consistency. The results of these tests reveal that this method is accurate, stable, and applicable to a wide range of atmospheric flows and scales.

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