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Journal ArticleDOI: 10.1080/03091929.2020.1774876

Thermal versus isothermal rotating shallow water equations: comparison of dynamical processes by simulations with a novel well-balanced central-upwind scheme

04 Mar 2021-Geophysical and Astrophysical Fluid Dynamics (Taylor & Francis)-Vol. 115, Iss: 2, pp 125-154
Abstract: We introduce a new high-resolution well-balanced central-upwind scheme for two-dimensional rotating shallow water equations with horizontal temperature/density gradients – thermal rotating shallow

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Topics: Shallow water equations (65%), Upwind scheme (62%)
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7 results found


Journal ArticleDOI: 10.1080/00107514.2018.1515254
Abstract: Fluid dynamics is a broad field within fluid mechanics that deals with the flow of fluids such as liquids and gases under various conditions, mostly with the dynamics of flows on macroscopic scales...

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Topics: Fluid mechanics (71%), Geophysical fluid dynamics (67%), Fluid dynamics (65%) ... read more

18 Citations


Open accessJournal ArticleDOI: 10.1051/M2AN/2021009
Alexander Kurganov1, Yongle Liu1, Yongle Liu2, Vladimir Zeitlin3  +1 moreInstitutions (3)
Abstract: We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x - and y -directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.

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Topics: Upwind scheme (58%), Dissipation (56%)

7 Citations


Open accessPosted Content
Abstract: Energy preserving reduced-order models are developed for the rotating thermal shallow water (RTSW) equation in the non-canonical Hamiltonian/Poisson form. The RTSW equation is discretized in space by the skew-symmetric finite-difference operators to preserve the Hamiltonian structure. The resulting system of ordinary differential equations is integrated in time by the energy preserving average vector field (AVF) method. An energy preserving, computationally efficient reduced-order model (ROM) is constructed by proper orthogonal decomposition (POD) with the Galerkin projection. The nonlinearities in the ROM are efficiently computed by discrete empirical interpolation method (DEIM). Preservation of the energy (Hamiltonian), and other conserved quantities; total mass, total buoyancy and total potential vorticity, by the reduced-order solutions are demonstrated which ensures the long term stability of the reduced-order solutions. The accuracy and computational efficiency of the ROMs are shown by a numerical test problem.

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1 Citations


Open accessJournal ArticleDOI: 10.1063/5.0064481
27 Oct 2021-Physics of Fluids
Abstract: In this paper, we show how the thermal effects affect trajectories, intensity, and formation of secondary structures during the passages of strong tropical cyclone-like vortices over oceanic warm and cold pools as well as over an island-type topography. Our results are obtained using the moist-convective thermal rotating shallow-water atmospheric model recently developed in [A. Kurganov et al., “Moist-convective thermal rotating shallow-water model,” Phys. Fluids 32, 066601 (2020)]. This model introduces thermodynamics of the moist air and moist convection in the standard rotating shallow-water models and allows to include in the latter atmosphere–ocean interactions in an elementary way.

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Topics: Atmospheric model (56%), Sea surface temperature (53%), Thermal (51%)

Open accessJournal ArticleDOI: 10.1007/S10915-021-01680-Z
Alina Chertock1, Alexander Kurganov2, Xin Liu3, Yongle Liu2  +2 moreInstitutions (5)
Abstract: We study the flux globalization based central-upwind scheme from Cheng et al. (J Sci Comput 80:538–554, 2019) for the Saint-Venant system of shallow water equations. We first show that while the scheme is capable of preserving moving-water equilibria, it fails to preserve much simpler “lake-at-rest” steady states. We then modify the computation of the global flux variable and develop a well-balanced scheme, which can accurately handle both still- and moving-water equilibria. In addition, we extend the flux globalization based central-upwind scheme to the case when “dry” and/or “almost dry” areas are present. To this end, we introduce a hybrid approach: we use the flux globalization based scheme inside the “wet” areas only, while elsewhere we apply the central-upwind scheme from Bollermann et al. (J Sci Comput 56:267–290, 2013), which is designed to accurately capture wet/dry fronts. We illustrate the performance of the proposed schemes on a number of numerical examples.

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Topics: Shallow water equations (55%), Flux (50%)

References
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48 results found


Journal ArticleDOI: 10.1016/0021-9991(79)90145-1
Abstract: A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”

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Topics: Godunov's scheme (71%), Godunov's theorem (63%), MUSCL scheme (59%) ... read more

6,147 Citations


Journal ArticleDOI: 10.1002/QJ.49710644905
A.E. Gill1Institutions (1)
Abstract: A simple analytic model is constructed to elucidate some basic features of the response of the tropical atmosphere to diabatic heating. In particular, there is considerable east-west asymmetry which can be illustrated by solutions for heating concentrated in an area of finite extent. This is of more than academic interest because heating in practice tends to be concentrated in specific areas. For instance, a model with heating symmetric about the equator at Indonesian longitudes produces low-level easterly flow over the Pacific through propagation of Kelvin waves into the region. It also produces low-level westerly inflow over the Indian Ocean (but in a smaller region) because planetary waves propagate there. In the heating region itself the low-level flow is away from the equator as required by the vorticity equation. The return flow toward the equator is farther west because of planetary wave propagation, and so cyclonic flow is obtained around lows which form on the western margins of the heating zone. Another model solution with the heating displaced north of the equator provides a flow similar to the monsoon circulation of July and a simple model solution can also be found for heating concentrated along an inter-tropical convergence line.

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Topics: Equatorial waves (62%), Thermal equator (60%), Equator (58%) ... read more

3,373 Citations


Journal ArticleDOI: 10.1137/0721062
Abstract: The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is expl...

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Topics: Flux limiter (71%), Total variation diminishing (56%), High-resolution scheme (54%) ... read more

2,310 Citations


Journal ArticleDOI: 10.1137/S003614450036757X
01 Jan 2001-Siam Review
Abstract: In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

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1,892 Citations


Journal ArticleDOI: 10.1006/JCPH.2000.6459
Alexander Kurganov1, Eitan Tadmor2Institutions (2)
Abstract: Central schemes may serve as universal finite-difference methods for solving nonlinear convection?diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax?Friedrichs scheme (P. D. Lax, 1954) is the forerunner for such central schemes. The central Nessyahu?Tadmor (NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these central schemes is of order O((?x)2r/?t). In the convective regime where ?t~?x, the improved resolution of the NT scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of central schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms.In this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O(?x)2r?1)). In particular, our new central schemes maintain their high-resolution independent of O(1/?t), and letting ?t ? 0, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these central schemes is the use of more precise information of the local propagation speeds. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these central schemes realize the (nonsmooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size.The semi-discrete central scheme is then extended to multidimensional problems, with or without degenerate diffusive terms. Fully discrete versions are obtained with Runge?Kutta solvers. We prove that a scalar version of our high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and nonconvex problems with and without degenerate diffusion, and scalar and systems of equations in one- and two-space dimensions. Time evolution is carried out by the third- and fourth-order explicit embedded integration Runge?Kutta methods recently proposed by A. Medovikov (1998). These numerical studies demonstrate the remarkable resolution of our new family of central scheme.

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Topics: Maximum principle (51%), Convection–diffusion equation (51%), Conservation law (51%) ... read more

1,486 Citations


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