An Optimized Implicit Scheme for Compressible Reactive Gas Flow

Dietrich, Daniel L.; Wormeck, John J.

doi:10.1080/01495728508961858

Cited by 7 publications

(2 citation statements)

References 7 publications

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“…The leapfrog scheme possesses a 2Dt computational mode, which can cause the numerical solution at even and odd time steps to split apart unphysically (e.g., Lilly 1965;Young 1968;Mesinger and Arakawa 1976). Various methods have been proposed to control the timesplitting instability (e.g., Kurihara 1965;Magazenkov 1980;Dietrich and Wormeck 1985;Roache and Dietrich 1988), the most common being to apply the (1, 22, 1) filter conceived by Robert (1966) and analyzed by Asselin (1972). The properties of the Robert-Asselin filter have been studied widely (e.g., Schlesinger et al 1983;D equ e and Cariolle 1986;Tandon 1987;Robert and L epine 1997;Cordero and Staniforth 2004;Marsaleix et al 2012) and the filter is used extensively in current models (e.g., Griffies et al 2001;Bartello 2002;Fraedrich et al 2005;Hartogh et al 2005;Williams et al 2009).…”

Section: Introductionmentioning

confidence: 98%

Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

Williams

2013

Monthly Weather Review

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The leapfrog time-stepping scheme is commonly used in general circulation models of weather and climate. The Robert-Asselin filter is used in conjunction with it, to damp the computational mode. Although the leapfrog scheme makes no amplitude errors when integrating linear oscillations, the Robert-Asselin filter introduces first-order amplitude errors. The RAW filter, which was recently proposed as an improvement, eliminates the first-order amplitude errors and yields third-order amplitude accuracy. This development has been shown to significantly increase the skill of medium-range weather forecasts. However, it has not previously been shown how to further improve the accuracy by eliminating the third-and higher-order amplitude errors. This presentation will show that leapfrogging over a suitably weighted blend of the filtered and unfiltered tendencies eliminates the third-order amplitude errors and yields fifth-order amplitude accuracy. It will also show that the use of a more discriminating (1, −4, 6, −4, 1) filter instead of a (1, −2, 1) filter eliminates the fifth-order amplitude errors and yields seventh-order amplitude accuracy. Other related schemes are obtained by varying the values of the filter parameters, and it is found that several combinations offer an appealing compromise of stability and accuracy. The proposed new schemes are shown to yield substantial forecast improvements in a medium-complexity atmospheric general circulation model. They appear to be attractive alternatives to the filtered leapfrog schemes currently used in many weather and climate models.

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Section: Introductionmentioning

confidence: 98%

Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

Williams

2013

Monthly Weather Review

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“…The leapfrog scheme exhibits a wellknown spurious computational mode (e.g., Lilly 1965;Young 1968;Mesinger and Arakawa 1976;Haltiner and Williams 1980;Durran 1999;Kalnay 2003), which may grow unphysically in nonlinear integrations. Unfortunately, many proposed strategies for controlling the computational mode, including the filtered leapfrogtrapezoidal scheme (Dietrich and Wormeck 1985) and the weighted filtered leapfrog-trapezoidal scheme (Roache and Dietrich 1988), incur a reduction in formal accuracy.…”

Section: Introductionmentioning

confidence: 99%

The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations

Williams

2011

Monthly Weather Review

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Errors caused by discrete time stepping may be an important component of total model error in contemporary atmospheric and oceanic simulations. To reduce time-stepping errors in leapfrog integrations, the Robert-Asselin-Williams (RAW) filter was proposed by the author as a simple improvement to the widely used Robert-Asselin (RA) filter. The present paper examines the behavior of the RAW filter in semi-implicit integrations. First, in a linear theoretical analysis, the stability and accuracy are interrogated by deriving analytic expressions for the amplitude errors and phase errors. Then, power-series expansions are used to interpret the leading-order errors for small time steps and hence to identify optimal values of the filter parameters. Finally, the RAW filter is tested in a realistic nonlinear setting, by applying it to semi-implicit integrations of the elastic pendulum equations. The results suggest that replacing the RA filter with the RAW filter could reduce time-stepping errors in semi-implicit integrations.

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An ocean modelling system with turbulent boundary layers and topography: numerical description

Dietrich

Marietta

Roache

1987

Numerical Methods in Fluids

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SUMMARYThe Sandia ocean modelling system (SOMS) is a system of three-dimensional, fully conservative, partially implicit numerical models based on primitive equations and a staggered Arakawa 'c' grid. A thin-shell bottom boundary layer submodel coupled to a free-stream submodel resolves boundary layers together with realistic topography. Both submodels use stretched vertical co-ordinates and an optional MellorYamada level-2.5 turbulence closure. Rigid top pressures are determined by vertical integration of the conservation equations using a hydrostatic approximation. SOMS reproduces previously published results, but with notable advantages in speed and economy.

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An Optimized Implicit Scheme for Compressible Reactive Gas Flow

Cited by 7 publications

References 7 publications

Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations

An ocean modelling system with turbulent boundary layers and topography: numerical description

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