The authors benefited from discussions with Frank Giraldo and from comments by the anonymous reviewers. This research was supported by the National Science Foundation through Grant ATM-0836316 and the Office of Naval Research through Contract N00173-10-1-G033.
Ascher, U. M., , S. J. Ruuth, , and B. T. R. Wetton, 1995: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal., 32, 797–823.
Benoit, R., , M. Desgagné, , P. Pellerin, , S. Pellerin, , Y. Chartier, , and S. Desjardins, 1997: The Canadian MC2: A semi-Lagrangian, semi-implicit wideband atmospheric model suited for finescale process studies and simulation. Mon. Wea. Rev., 125, 2382–2415.
Durran, D. R., 1991: The third-order Adams–Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720.
Durran, D. R., 2008: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech., 601, 365–379.
Evans, K. J., , M. A. Taylor, , and J. B. Drake, 2010: Accuracy analysis of a spectral element atmospheric model using a fully implicit solution framework. Mon. Wea. Rev., 138, 3333–3341.
Fornberg, B., , and T. A. Driscoll, 1999: A fast spectral algorithm for nonlinear wave equations with linear dispersion. J. Comput. Phys., 155, 456–467.
Frank, J., , W. Hundsdorfer, , and J. G. Verwer, 1997: On the stability of implicit-explicit linear multistep methods. Appl. Numer. Math., 25, 193–205.
Giraldo, F. X., 2005: Semi-implicit time-integrators for a scalable spectral element atmospheric model. Quart. J. Roy. Meteor. Soc., 131, 2431–2454, doi:10.1256/qj.03.218.
Giraldo, F. X., , M. Restelli, , and M. Läuter, 2010: Semi-implicit formulations of the Navier–Stokes equations: Application to nonhydrostatic atmospheric modeling. SIAM J. Sci. Comput., 32, 3394–3425.
Karniadakis, G. E., , M. Israeli, , and S. A. Orszag, 1991: High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys., 97, 414–443.
Kennedy, C. A., , and M. H. Carpenter, 2003: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math., 44, 139–181, doi:10.1016/S0168-9274(02)00138-1.
Klemp, J. B., , and R. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35, 1070–1096.
Kwizak, M., , and A. J. Robert, 1971: A semi-implicit scheme for grid point atmospheric models of the primitive equation. Mon. Wea. Rev., 99, 32–36.
Lipps, F., , and R. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39, 2192–2210.
Nevanlinna, O., , and W. Liniger, 1978: Contractive methods for stiff differential equations Part I. BIT Numer. Math., 18, 457–474.
Ogura, Y., , and N. A. Phillips, 1962: Scale analysis for deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173–179.
Robert, A. J., 1966: The integration of a low order spectral form of the primitive meteorological equations. J. Meteor. Soc. Japan, 44, 237–244.
Ullrich, P., , and C. Jablonowski, 2012: Operator-split Runge–Kutta–Rosenbrock methods for nonhydrostatic atmospheric models. Mon. Wea. Rev., 140, 1257–1284.
Varah, J. M., 1980: Stability restrictions on second order, three level finite difference schemes for parabolic equations. SIAM J. Numer. Anal., 17, 300–309.
Wicker, L. J., , and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097.
Williams, P. D., 2011: The RAW filter: An improvement to the Robert–Asselin filter in semi-implicit integrations. Mon. Wea. Rev., 139, 1996–2007.
An N-step linear multistep method has N amplification factors, one of which corresponds to the physical mode. The others are associated with computational modes. Unless otherwise specified, |A| will refer to the maximum of the magnitude of the amplification factor over all physical and computational modes.
T1, the standard trapezoidal method, in which the time difference is computed over an interval of Δt rather than 2Δt as in (4), would give the lowest phase error.
The case ϵ = 0 is not included because for many schemes that are unstable arbitrarily close to the origin, |A| → 1 as |ωHΔt| → 0.
For example, if N = 0.01 s−1 and cs = 300 m s−1, N2/cs2 can be neglected in comparison with l2 unless the vertical wavelength exceeds 100 km.
Throughout the following analysis of the T2θ–LF scheme, we assume θ = 0.5 and no time filtering.