A modified leapfrog scheme for shallow water equations

Sun, Wen‐Yih; Sun, Oliver

doi:10.1016/j.compfluid.2011.08.019

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…It can be shown that this stability condition decreases to α igw = 1 on a staggered grid (e.g. Sun and Sun, 2011). accurate representation, good stability properties and ease of implementation of the different terms involved in the equations, as discussed in Section 2.1.…”

Section: Internal Gravity Wavesmentioning

confidence: 97%

Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations

et al. 2015

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International audienceExcept for vertical diffusion (and possibly the external mode and bottom drag), oceanic models usually rely on explicit time-stepping algorithms subject to Courant-Friedrichs-Lewy (CFL) stability criteria. Implicit methods could be unconditionally stable, but an algebraic system must be solved at each time step and other considerations such as accuracy and efficiency are less straightforward to achieve. Depending on the target application, the process limiting the maximum allowed time-step is generally different. In this paper, we introduce offline diagnostics to predict stability limits associated with internal gravity waves, advection, diffusion, and rotation. This suite of diagnostics is applied to a set of global, regional and coastal numerical simulations with several horizontal/vertical resolutions and different numerical models. We show that, for resolutions finer that 1/2◦, models with an eulerian vertical coordinate are generally constrained by vertical advection in a few hot spots and that numerics must be extremely robust to changes in Courant number. Based on those results, we review the stability and accuracy of existing numerical kernels in vogue in primitive equations oceanic models with a focus on advective processes and the dynamics of internal waves. We emphasize the additional value of studying the numerical kernel of oceanic models in the light of coupled space-time approaches instead of studying the time schemes independently from spatial discretizations. From this study, we suggest some guidelines for the development of temporal schemes in future generation multi-purpose oceanic models

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Section: Internal Gravity Wavesmentioning

confidence: 97%

Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations

et al. 2015

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“…Table 1. Ratio of computed phase speed to analytic phase speed C ph /C as function of wavelength (λ) and Courant number Co for 2nd-order scheme according to (24) [37].…”

Section: Modified Leapfrog Scheme For Shallow Water Equationsmentioning

confidence: 99%

“…At the crest, the simulated velocity s c U is identical to * s U , which has a physical solution (i.e., a real number instead of a complex number) of Equation (41). Equation (37) is also held at the crest, but not at the jump at x = 627 with u =175.7 cm s −1 , where the velocity calculated from Equation ( 41) is only 157.58 cm s −1 . The Bernoulli function (black line) drops from 22,948 to 20,485 cm 2 s −2 across the jump at t = 96 s, but J remains almost the same (284,582 vs. 284,581 cm 3 s −2 ), except small kinks at the jump and obstacle foots, because it is derived from an inviscid Equation (38) of mass flux in Figure 9.…”

Section: Hydraulic Jumpsmentioning

confidence: 99%

Challenges and Progress in Computational Geophysical Fluid Dynamics in Recent Decades

Sun

2023

Atmosphere

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Here we present the numerical methods, applications, and comparisons with observations and previous studies. It includes numerical analyses of shallow water equations, Sun’s scheme, and nonlinear model simulations of a dam break, solitary Rossby wave, and hydraulic jump without smoothing. We reproduce the longitude and transverse cloud bands in the Equator; two-day mesoscale waves in Brazil; Ekman spirals in the atmosphere and oceans, and a resonance instability at 30° from the linearized equations. The Purdue Regional Climate Model (PRCM) reproduces the explosive severe winter storms in the Western USA; lee-vortices in Taiwan; deformation of the cold front by mountains in Taiwan; flooding and drought in the USA; flooding in Asia; and the Southeast Asia monsoons. The model can correct the small-scale errors if the synoptic systems are correct. Usually, large-scale systems are more important than small-scale disturbances, and the predictability of NWP is better than the simplified dynamics models. We discuss the difference between Boussinesq fluid and the compressible fluid. The Bernoulli function in compressible atmosphere conserving the total energy, is better than the convective available potential energy (CAPE) or the Froude number, because storms can develop without CAPE, and downslope wind can form against a positive buoyancy. We also present a new terrain following coordinate, a turbulence-diffusion model in the convective boundary layer (CBL), and a new backward-integration model including turbulence mixing in the atmosphere.

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“…The eigenvalues and non-linear simulations of the nonhydrostatic model (NH) using the conventional forwardÁ backward scheme (FB) (Mesinger and Arakawa, 1976;Sun, 1980Sun, , 1984, the horizontal explicit and vertical implicit scheme (HEÁVI) (Klemp and Wilhelmson, 1978;Saito, 2007) and the modified non-hydrostatic model (MNH) have been discussed in Sun et al (2012), hereafter referred to as Part I. The MNH is designed to suppress highfrequency acoustic waves by multiplying the left-hand side of the continuity equation by a parameter d, where 16 ]d ]4.…”

Section: Introductionmentioning

confidence: 99%

“…

A B S T R A C T Sun et al (2012) proposed a modified non-hydrostatic model (MNH), in which the left-hand side of the continuity equation is multiplied by a parameter d (45d516 in the article) to suppress high-frequency acoustic waves. They showed that the MNH allows a longer time step than the original non-hydrostatic model (NH).

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mentioning

confidence: 99%

A modified atmospheric non-hydrostatic model on low aspect ratio grids: part II

Sun

Tsuboki

2013

Tellus A: Dynamic Meteorology and Oceanography

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A B S T R A C T Sun et al. (2012) proposed a modified non-hydrostatic model (MNH), in which the left-hand side of the continuity equation is multiplied by a parameter d (45d516 in the article) to suppress high-frequency acoustic waves. They showed that the MNH allows a longer time step than the original non-hydrostatic model (NH). The MNH is also more accurate and efficient than the horizontal explicit and vertical implicit scheme (HEÁVI) when the aspect ratio (Dx/Dz) is small. In addition to multiplying a parameter d, here we propose to add a smoothing on the right-hand side of the continuity equation in the MNH to damp shortest sound waves. Linear stability analysis and non-linear model simulations show that the MNH with smoothing (henceforth abbreviated as MNHS) can use twice the time interval of the MNH while maintaining the same accuracy. The MNHS is also more accurate and efficient than HEÁVI when the aspect ratio is small.

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A modified leapfrog scheme for shallow water equations

Cited by 7 publications

References 12 publications

Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations

Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations

Challenges and Progress in Computational Geophysical Fluid Dynamics in Recent Decades

A modified atmospheric non-hydrostatic model on low aspect ratio grids: part II

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