High Order Well-Balanced Schemes Based on Numerical Reconstruction of the Equilibrium Variables

Russo, Giovanni; Khe, A. K.

doi:10.1142/9789814317429_0032

Cited by 13 publications

(12 citation statements)

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“…More precisely, S = κ exp(η/c v ), where κ is a constant (in classical thermodynamics the entropy is defined up to an additive constant), and c v denotes the specific heat at constant volume. Stationary solutions are therefore characterized by three invariants, Q, h, and S, expressed by relations (34,36,39). We shall make use of this property as a key ingredient for the construction of well balanced schemes.…”

Section: Equilibriamentioning

confidence: 99%

Linearly implicit all Mach number shock capturing schemes for the Euler equations

Avgerinos

Bernard

Iollo

et al. 2019

Journal of Computational Physics

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We propose a family of simple second order accurate schemes for the numerical solution of Euler equation of gas dynamics that are (linearly) implicit in the acoustic waves, eliminating the acoustic CFL restriction on the time step. The general idea is that explicit differential operators in space relative to convective or material speeds are discretized by upwind schemes or local Lax-Friedrics fluxes and the linear implicit operators, pertaining to acoustic waves, are discretized by central differences. We have compared the results of such schemes on a series of one-dimensional test problems including classical shock tube configurations. Also we have considered low-Mach number acoustic wave propagation tests as well as nozzle flows in various Mach regimes. The results show that these schemes do not introduce excessive numerical dissipation at low Mach number providing an accurate solution in such regimes. They perform reasonably well also when the Mach number are not too small.

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Section: Equilibriamentioning

confidence: 99%

Linearly implicit all Mach number shock capturing schemes for the Euler equations

Avgerinos

Bernard

Iollo

et al. 2019

Journal of Computational Physics

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“…Since we have imposed a smooth topography, in the present work we deal with smooth steady states. The reader is referred to [13,33,38,39] where extension to discontinuous buttom is considered that makes discontinuous the steady water height (see also [28,29] to related studies).…”

Section: Characteristic Fields and Riemann Invariants Easy Calculatimentioning

confidence: 99%

“…During the last two decades, after the works by Bermudez-Vasquez [2] and Greenberg-LeRoux [24] (see also [21,22,23]), the derivation of well-balanced schemes able to restore the lake at rest (1.4) was a very active research topic. Several strategies have been derived (for instance see [1,30,26,10,21,24,11,32,31,34,33,12,38,39]). The main difficulty coming from the derivation of wellbalanced schemes remains in the discretization of the topography source term to be consistent with the lake at rest.…”

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confidence: 99%

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

Berthon

Chalons

2015

Math. Comp.

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This work is devoted to the derivation of a fully well-balanced numerical scheme for the well-known shallow-water model. During the last two decades, several well-balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so-called lake at rest. By fully well-balanced, we mean here that the proposed Godunov-type method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality.

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“…A good numerical method for the system (1.1) should thus be well-balanced (in the sense that it must exactly preserve physically relevant steady states) and positivity preserving (in the sense that the computed values of h must be positive). In the past two decades, many well-balanced schemes have been developed (e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Some of them preserve only 'lake at rest' steady states, that is, u Á v Á 0, h C B Á constant, [2-10, 12, 13], other can preserve a nonflat steady-state solution as well, [11,[15][16][17].…”

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confidence: 99%

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Chertock

Cui

Kurganov

et al. 2015

Numerical Methods in Fluids

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Shallow water models are widely used to describe and study free-surface water flow. Even though, in many practical applications the bottom friction does not have much influence on the solutions, the friction terms will play a significant role when the depth of the water is very small. In this paper, we study the Saint-Venant system of shallow water equations with friction terms and develop a well-balanced central-upwind scheme that is capable of exactly preserving its steady states. The scheme also preserves the positivity of the water depth. We test the designed scheme on a number of one-and two-dimensional examples that demonstrate robustness and high resolution of the proposed numerical approach. The data in the last numerical example correspond to the laboratory experiments reported in [L. Cea, M. Garrido, and J. Puertas, J. Hydrol, 382 (2010), pp. 88-102], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special technique, which helps to achieve a remarkable agreement between the numerical and experimental results.

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High Order Well-Balanced Schemes Based on Numerical Reconstruction of the Equilibrium Variables

Cited by 13 publications

References 0 publications

Linearly implicit all Mach number shock capturing schemes for the Euler equations

Linearly implicit all Mach number shock capturing schemes for the Euler equations

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

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