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A Semi-Implicit Modification to the Lorenz N-Cycle Scheme and Its JUNE 2016 H O T T A E T A L . 2215 A Semi-Implicit Modification to the Lorenz N-Cycle Scheme and Its Application for Integration of Meteorological Equations DAISUKE HOTTA Japan Meteorological Agency, Tokyo, Japan, and University of Maryland, College Park, College Park, Maryland EUGENIA KALNAY University of Maryland, College Park, College Park, Maryland PAUL ULLRICH Department of Land, Air and Water Resources, University of California, Davis, Davis, California (Manuscript received 23 September 2015, in final form 13 January 2016) ABSTRACT The Lorenz N-cycle is an economical time integration scheme that requires only one function evaluation per time step and a minimal memory footprint, but yet possesses a high order of accuracy. Despite these advantages, it has remained less commonly used in meteorological applications, partly because of its lack of semi-implicit formulation. In this paper, a novel semi-implicit modification to the Lorenz N-cycle is proposed. The advantage of the proposed new scheme is that it preserves the economical memory use of the original explicit scheme. Unlike the traditional Robert–Asselin (RA) filtered semi-implicit leapfrog scheme whose formal accuracy is only of first order, the new scheme has second-order accuracy if it adopts the Crank– Nicolson scheme for the implicit part. A linear stability analysis based on a univariate split-frequency os- cillation equation suggests that the 4-cycle is more stable than other choices of N. Numerical experiments performed using the dynamical core of the Simplified Parameterizations Primitive Equation Dynamics (SPEEDY) atmospheric general circulation model under the framework of the Jablonowski–Williamson baroclinic wave test case confirms that the new scheme in fact has second-order accuracy and is more accurate than the traditional RA-filtered leapfrog scheme. The experiments also give evidence for Lorenz’s claim that the explicit 4-cycle scheme can be improved by running its two ‘‘isomeric’’ versions in alternating sequences. Unlike the explicit scheme, however, the proposed semi-implicit scheme is not improved by alternation of the two versions. 1. Introduction (PDEs) that describe the governing laws of geophysical fluid flows. There is thus a high demand for improvements A unique feature of the atmospheric and oceanic of accuracy of such models. sciences is that, unlike other fields of natural sciences, One of the major challenges in designing numerical controlled experiments are difficult to perform. Accord- integration schemes for atmospheric models, in par- ingly, numerical experimentation has become an increas- ticular the atmospheric general circulation models ingly important methodology in meteorology and physical (AGCMs), is the so-called stiffness problem: the equa- oceanography. A key role in numerical experimentation is tions solved by AGCMs contain not only the slower played by atmospheric or oceanic models which numeri- waves that are relevant to the actual weather phenom- cally integrate hydrodynamic partial differential equations ena, but also the faster waves that are of little meteo- rological interest. The phase speeds of the faster waves are typically an order-of-magnitude faster than those of Corresponding author address: Daisuke Hotta, Numerical Pre- diction Division, Japan Meteorological Agency, 1-3-4 Otemachi, the slower waves. To satisfy the Courant–Friedrichs– Chiyoda-ku, Tokyo 100-8122, Japan. Lewy (CFL) stability condition, an explicit temporal E-mail: [email protected] integration scheme needs to use an overly short time DOI: 10.1175/MWR-D-15-0330.1 Ó 2016 American Meteorological Society Unauthenticated | Downloaded 10/01/21 06:06 AM UTC 2216 MONTHLY WEATHER REVIEW VOLUME 144 step just to maintain stability, making the integration issue is commonly dealt with by running a two-time- significantly more expensive. While several advanced level scheme, such as forward-Euler scheme, at the very approaches have been proposed that resolve this issue, first step. The third issue can be resolved by filtering out including Laplace transform methods (Clancy and the computational mode by applying the Robert– Lynch 2011), semi-implicit predictor-corrector methods Asselin (RA) filter (Robert 1966; Asselin 1972); this (Clancy and Pudykiewicz 2013a), and exponential in- treatment also introduces the side effect of degrading tegrator methods (Clancy and Pudykiewicz 2013b), the the scheme to first-order accuracy by damping not only solution adopted by most current AGCMs is to use a the computational mode but also the physical mode. semi-implicit scheme that treats the terms responsible Despite these disadvantages, the leapfrog scheme with for the fast waves implicitly and the other terms ex- semi-implicit modification (Robert 1969), combined plicitly (Robert 1969); this treatment clears the CFL with RA filter and separate treatment of dissipative part, condition for the fast waves and thus allows an efficient has been the most widely used scheme for AGCMs, integration with much longer time steps. The availability and a better scheme free from these limitations has been of this semi-implicit treatment has been an important long sought. factor for a temporal integration scheme to be used in One way to achieve this goal is to alleviate the AGCMs. It should be noted, however, that the advan- limitations by improving the classical RA-filtered tage of the semi-implicit approach is challenged by the leapfrog scheme. Recently, Williams (2009) pro- increasing trend in high-performance computing to ex- posed an improvement to the RA filter. The new filter, ploit massively parallel machines with relatively slow called Robert–Asselin–Williams (RAW) filter, pre- internode communications because implicit solvers typi- serves the second-order accuracy of the leapfrog cally necessitate global communication that risks making scheme without any significant increase in computa- the scheme less scalable. We shall revisit this point in the tional cost. The advantage of the RAW filter over the last section of the paper. RA filter is confirmed also for the semi-implicit leap- Traditionally, in the community of atmospheric mod- frog scheme (Williams 2011; Amezcua et al. 2011). eling, a rather simple, three-step centered-differencing Williams (2013) devised further improvements in this scheme, commonly known as the leapfrog scheme, has line, leading to schemes with even higher accuracy in been regarded as the method of choice and adopted by amplitude (up to seventh order; the phase error re- many AGCMs. In pursuit of more accuracy and sophis- mains second order). While these improved schemes tication, recent new-generation AGCMs have switched effectively eliminate the undesirable artificial damping to more advanced higher-order temporal integration of physical modes, other shortcomings of the filtered schemes. At present, however, the leapfrog is still in leapfrog scheme (i.e., the instability for dissipative wide use in many traditional AGCMs, perhaps because terms and the necessity of special treatment for the of its desirable properties, which include the following: initial steps) remain unresolved. The efficacy of ren- ease of implementation, availability of a stable semi- dering the RA filter’s second order is also diminished implicit version, low cost in computational time, low if a first-order scheme is used for dissipative terms to memory consumption, and conservation of energy for a suppress instability. nondissipative system. Attempts have also been made to seek for alternative The above desirable properties are, however, tainted schemes that are better suited for atmospheric and by the following undesirable features (Durran 1991). oceanic models. Multistep schemes such as the Adams– First, the scheme is unstable when applied to a system Bashforth family of schemes, for example, can have with dissipation. Second, being a three-time-level the order of accuracy that is higher than the leapfrog scheme, it necessitates special treatment at the very without increasing computational expenses. Durran first several steps. Last, and most importantly, the leap- (1991) found, however, that while the three-step frog scheme produces, when applied to a nonlinear third-order Adams–Bashforth scheme is a viable al- system, a spurious computational mode, which, if left ternative to the RA-filtered explicit leapfrog scheme, unattenuated, results in time-splitting instability. In this scheme cannot replace the semi-implicit leapfrog AGCMs, the first issue is typically dealt with by applying scheme because the Adams–Bashforth scheme be- the leapfrog only to nondissipative dynamics part; dis- comes unstable if it is combined with a semi-implicit sipative processes such as physics and damping are scheme for fast modes. The Runge–Kutta family of treated with separate schemes, such as the explicit schemes can also be more accurate than the leapfrog. forward-Euler or implicit backward-Euler scheme. This Kar (2006) and Whitaker and Kar (2013), for example, treatment unfortunately comes with the side effect of have successfully developed semi-implicit versions making the scheme only first-order accurate. The second of Runge–Kutta-type schemes and showed their Unauthenticated | Downloaded 10/01/21 06:06 AM UTC JUNE 2016 H O T T A E T A L . 2217 advantages over the RA-filtered semi-implicit leap- whoseresultsarepresentedinsection 5. Section 6 frog scheme. Their schemes, however, are more concludes the paper with a summary and an outlook computationally expensive and consume more mem- for future research. ory than
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