Editorial Type:
Article
Article Type:
Research Article

A Time-Splitting Scheme for the Elastic Equations Incorporating Second-Order Runge–Kutta Time Differencing

Louis J. Wicker Department of Meteorology, Texas A&M University, College Station, Texas

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William C. Skamarock National Center for Atmospheric Research,* Boulder, Colorado

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Print Publication:
01 Jul 1998
DOI:
https://doi.org/10.1175/1520-0493(1998)126<1992:ATSSFT>2.0.CO;2
Page(s):
1992–1999
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Abstract

A forward-in-time splitting method for integrating the elastic equations is presented. A second-order Runge–Kutta time integrator (RK2) for the large-time-step integration is combined with the forward–backward scheme in a manner similar to the Klemp and Wilhelmson method. The new scheme produces fully second-order-accurate integrations for advection and gravity wave propagation. The RK2 scheme uses upwind discretizations for the advection terms and is easily combined with standard vertically semi-implicit techniques so as to improve computational efficiency when the grid aspect ratio becomes large. A stability analysis of the RK2 split-explicit scheme shows that it is stable for a wide range of advective and acoustic wave Courant numbers.

The RK2 time-split scheme is used in a full-physics nonhydrostatic compressible cloud model. The implicit damping properties associated with the RK2’s third-order horizontal differencing allows for a significant reduction in the value of horizontal filtering applied to the momentum and pressure fields, while qualitatively the solutions appear to be better resolved than solutions from a leapfrog model.

Corresponding author address: Dr. Louis J. Wicker, 1204 Eller O&M Bldg., College Station, TX 77843-3150.

Abstract

A forward-in-time splitting method for integrating the elastic equations is presented. A second-order Runge–Kutta time integrator (RK2) for the large-time-step integration is combined with the forward–backward scheme in a manner similar to the Klemp and Wilhelmson method. The new scheme produces fully second-order-accurate integrations for advection and gravity wave propagation. The RK2 scheme uses upwind discretizations for the advection terms and is easily combined with standard vertically semi-implicit techniques so as to improve computational efficiency when the grid aspect ratio becomes large. A stability analysis of the RK2 split-explicit scheme shows that it is stable for a wide range of advective and acoustic wave Courant numbers.

The RK2 time-split scheme is used in a full-physics nonhydrostatic compressible cloud model. The implicit damping properties associated with the RK2’s third-order horizontal differencing allows for a significant reduction in the value of horizontal filtering applied to the momentum and pressure fields, while qualitatively the solutions appear to be better resolved than solutions from a leapfrog model.

Corresponding author address: Dr. Louis J. Wicker, 1204 Eller O&M Bldg., College Station, TX 77843-3150.

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