Editorial Type:
Article
Article Type:
Research Article

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

Paul Ullrich Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, Ann Arbor, Michigan

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Christiane Jablonowski Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, Ann Arbor, Michigan

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Print Publication:
01 Apr 2012
DOI:
https://doi.org/10.1175/MWR-D-10-05073.1
Page(s):
1257–1284
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Abstract

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 a traditional explicit Runge–Kutta formula in the horizontal is proposed. Up to third-order temporal accuracy 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.

Corresponding author address: Paul Ullrich, Department of Atmospheric, Oceanic and Space Sciences, Space Research Building, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109. E-mail: paullric@umich.edu

Abstract

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 a traditional explicit Runge–Kutta formula in the horizontal is proposed. Up to third-order temporal accuracy 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.

Corresponding author address: Paul Ullrich, Department of Atmospheric, Oceanic and Space Sciences, Space Research Building, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109. E-mail: paullric@umich.edu
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