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Alexander Kurganov - One of the best experts on this subject based on the ideXlab platform.
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an adaptive well balanced positivity preserving central Upwind Scheme on quadtree grids for shallow water equations
Computers & Fluids, 2020Co-Authors: Mohammad A Ghazizadeh, Abdolmajid Mohammadian, Alexander KurganovAbstract:Abstract We present an adaptive well-balanced positivity preserving central-Upwind Scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-Upwind Scheme proposed in [ A. Kurganov and G. Petrova , Commun. Math. Sci., 5 (2007), pp. 133–160]. The designed Scheme is well-balanced in the sense that it is capable of exactly preserving “lake-at-rest” steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed Scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-Upwind quadtree Scheme.
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a well balanced central Upwind Scheme for the thermal rotating shallow water equations
Journal of Computational Physics, 2020Co-Authors: Alexander Kurganov, Yongle Liu, V ZeitlinAbstract:Abstract We develop a well-balanced central-Upwind Scheme for rotating shallow water model with horizontal temperature and/or density gradients—the thermal rotating shallow water (TRSW). The Scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-Upwind Scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria. The designed Scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical Scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.
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adaptive moving mesh Upwind Scheme for the two species chemotaxis model
Computers & Mathematics With Applications, 2019Co-Authors: Alina Chertock, Alexander Kurganov, Mario RicchiutoAbstract:Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacteria/cells in response to an external stimulus, usually a chemical one. A common property of all chemotaxis systems is their ability to model a concentration phenomenon-rapid growth of the cell density in small neighborhoods of concentration points/curves. More precisely, the solution may develop singular, spiky structures, or even blow up in finite time. Therefore, the development of accurate and computationally efficient numerical methods for the chemotaxis models is a challenging task. We study the two-species Patlak-Keller-Segel type chemotaxis system, in which the two species do not compete, but have different chemotactic sensitivities, which may lead to a significantly difference in cell density growth rates. This phenomenon was numerically investigated in [Kurganov and Lukacova-Medvid'ova, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 131-152] and [Chertock et al., Adv. Comput. Math., 44 (2018), pp. 327-350], where second-and higher-order methods on uniform Cartesian grids were developed. However, in order to achieve high resolution of the density spikes developed by the species with a lower chemotactic sensitivity, a very fine mesh had to be utilized and thus the efficiency of the numerical method was affected. In this work, we consider an alternative approach relying on mesh adaptation, which helps to improve the approximation of the singular structures evolved by chemotaxis models. We develop, in particular, an adaptive moving mesh (AMM) finite-volume semi-discrete Upwind method for the two-species chemotaxis system. The proposed AMM technique allows one to increase the density of mesh nodes at the blowup regions. This helps to substantially improve the resolution while using a relatively small number of finite-volume cells.
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well balanced positivity preserving central Upwind Scheme with a novel wet dry reconstruction on triangular grids for the saint venant system
Journal of Computational Physics, 2018Co-Authors: Xin Liu, Alexander Kurganov, Yekaterina Epshteyn, Jason AlbrightAbstract:Abstract In this paper, we construct an improved well-balanced positivity preserving central-Upwind Scheme for the two-dimensional Saint-Venant system of shallow water equations. As in Bryson et al. (2011) [7] , our Scheme is based on a continuous piecewise linear discretization of the bottom topography over an unstructured triangular grid. The main new technique is a special reconstruction of the water surface in partially flooded cells. This reconstruction is an extension of the one-dimensional wet/dry reconstruction from Bollermann et al. (2013) [3] . The positivity of the computed water depth is enforced using the “draining” time-step technique introduced in Bollermann et al. (2011) [4] . The performance of the proposed central-Upwind Scheme is tested on a number of numerical experiments.
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second order fully discrete central Upwind Scheme for two dimensional hyperbolic systems of conservation laws
SIAM Journal on Scientific Computing, 2017Co-Authors: Alexander Kurganov, Martina PruggerAbstract:In this paper, we derive a new second-order fully discrete Godunov-type central-Upwind Scheme for two-dimensional hyperbolic systems of conservation laws. The Scheme is derived in three steps: reco...
Guergana Petrova - One of the best experts on this subject based on the ideXlab platform.
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central Upwind Scheme on triangular grids for the saint venant system of shallow water equations
NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, 2011Co-Authors: Steve Bryson, Alexander Kurganov, Yekaterina Epshteyn, Guergana PetrovaAbstract:We consider a novel second‐order central‐Upwind Scheme for the Saint‐Venant system of shallow water equations on triangular grids which was originally introduced in [3]. Here, in several numerical experiments we demonstrate accuracy, high resolution and robustness of the proposed method.
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well balanced positivity preserving central Upwind Scheme on triangular grids for the saint venant system
Mathematical Modelling and Numerical Analysis, 2011Co-Authors: Steve Bryson, Alexander Kurganov, Yekaterina Epshteyn, Guergana PetrovaAbstract:We introduce a new second-order central-Upwind Scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the Scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new Scheme, as well as its high resolution and robustness in a number of numerical examples.
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a central Upwind Scheme for nonlinear water waves generated by submarine landslides
2008Co-Authors: Alexander Kurganov, Guergana PetrovaAbstract:We study a simple one-dimensional (1-D) toy model for landslides-generated nonlinear water waves. The landslide is modeled as a rigid bump translating down the side of the bottom while the water motion is modeled by the Saint-Venant system of shallow water equations. The resulting system is numerically solved using a well-balanced positivity preserving central-Upwind Scheme. The obtained numerical results are in good agreement with both the two-dimensional (2-D) incompressible flow numerical simulations and the experimental data.
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a second order well balanced positivity preserving central Upwind Scheme for the saint venant system
Communications in Mathematical Sciences, 2007Co-Authors: Alexander Kurganov, Guergana PetrovaAbstract:A family of Godunov-type central-Upwind Schemes for the Saint-Venant system of shallow water equations has been first introduced in (A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002). Depending on the reconstruction step, the second-order versions of the Schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved second-order central-Upwind Scheme which, unlike its forerun- ners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed Scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new Scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.
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Compressible two-phase flows by central and Upwind Schemes
ESAIM: Mathematical Modelling and Numerical Analysis, 2004Co-Authors: Smadar Karni, Alexander Kurganov, Eduard-wilhelm Kirr, Guergana PetrovaAbstract:This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central Scheme and a Roe-type Upwind Scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed. Mathematics Subject Classification. 35L65, 65M06, 76N15, 76T99.
Abdelaziz Beljadid - One of the best experts on this subject based on the ideXlab platform.
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a central Upwind Scheme with artificial viscosity for shallow water flows in channels
Advances in Water Resources, 2016Co-Authors: Gerardo Hernandezduenas, Abdelaziz BeljadidAbstract:We develop a new high-resolution, non-oscillatory semi-discrete central-Upwind Scheme with artificial viscosity for shallow-water flows in channels with arbitrary geometry and variable topography. The artificial viscosity, proposed as an alternative to nonlinear limiters, allows us to use high-resolution reconstructions at a low computational cost. The Scheme recognizes steady states at rest when a delicate balance between the source terms and flux gradients occurs. This balance in irregular geometries is more complex than that taking place in channels with vertical walls. A suitable technique is applied by properly taking into account the effects induced by the geometry. Incorporating the contributions of the artificial viscosity and an appropriate time step restriction, the Scheme preserves the positivity of the water’s depth. A description of the proposed Scheme, its main properties as well as the proofs of well-balance and the positivity of the Scheme are provided. Our numerical experiments confirm stability, well-balance, positivity-preserving properties and high resolution of the proposed method. Comparisons of numerical solutions obtained with the proposed Scheme and experimental data are conducted, showing a good agreement. This Scheme can be applied to shallow-water flows in channels with complex geometry and variable bed topography.
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Well-balanced positivity preserving cell-vertex central-Upwind Scheme for shallow water flows
Computers & Fluids, 2016Co-Authors: Abdelaziz Beljadid, Abdolmajid Mohammadian, Alexander KurganovAbstract:Abstract We develop a new second-order two-dimensional central-Upwind Scheme on cell-vertex grids for approximating solutions of the Saint-Venant system with source terms due to bottom topography. Central-Upwind Schemes are developed based on the information about the local speeds of wave propagation. Compared to the triangular central-Upwind Schemes, the proposed cell-vertex one has an advantage of using more cell interfaces which provide more information on the waves propagating in different directions. We propose a new piecewise linear approximation of the bottom topography and a novel non-oscillatory reconstruction in which the gradient of each variable is computed using a modified minmod-type method to ensure the stability of the Scheme. A new technique is proposed for the correction of the water surface elevation which guarantees the positivity of the water depth. The well-balanced property of the proposed central-Upwind Scheme is ensured using a special discretization for the cell averages of the topography source terms. The proposed Scheme is tested on a number of numerical examples, among which we consider steady-state solutions with almost dry areas and their perturbations and solutions with rapidly varying flows over discontinuous bottom topography. Our numerical experiments confirm stability, well-balanced, positivity preserving properties and second-order accuracy of the proposed method. This Scheme can be applied to shallow water models when the bed topography is discontinuous and/or highly oscillatory, and on complicated domains where the use of unstructured grids is advantageous.
Abdolmajid Mohammadian - One of the best experts on this subject based on the ideXlab platform.
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an adaptive well balanced positivity preserving central Upwind Scheme on quadtree grids for shallow water equations
Computers & Fluids, 2020Co-Authors: Mohammad A Ghazizadeh, Abdolmajid Mohammadian, Alexander KurganovAbstract:Abstract We present an adaptive well-balanced positivity preserving central-Upwind Scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-Upwind Scheme proposed in [ A. Kurganov and G. Petrova , Commun. Math. Sci., 5 (2007), pp. 133–160]. The designed Scheme is well-balanced in the sense that it is capable of exactly preserving “lake-at-rest” steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed Scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-Upwind quadtree Scheme.
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extension of a well balanced central Upwind Scheme for variable density shallow water flow equations on triangular grids
Computers & Fluids, 2017Co-Authors: Sepideh Khorshid, Abdolmajid Mohammadian, Ioan NistorAbstract:Abstract In this paper, the central Upwind Scheme for variable density shallow water system of equations is extended to triangular discretization of the domain. In this Scheme, the well-balanced and positivity preserving properties are maintained such that the large oscillations and noises are avoided in the solution. Furthermore, time-history of flow surface always remains non-negative throughout the simulations. Various properties of the Scheme are validated using several benchmark data. Also, the accuracy and efficiency of the methodology are tested by comparing the results of the model to other complex Scheme for some test cases. The method ensures high computational efficiency while maintaining the accuracy of the results and preserves two types of “lake at rest” steady states, and is oscillation free across the small density change.
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a well balanced positivity preserving central Upwind Scheme for shallow water equations on unstructured quadrilateral grids
Computers & Fluids, 2016Co-Authors: Hamidreza Shirkhani, Abdolmajid Mohammadian, Ousmane Seidou, Alexander KurganovAbstract:Abstract We introduce a new second-order central-Upwind Scheme for shallow water equations on the unstructured quadrilateral grids. We propose a new technique for bottom topography approximation over quadrilateral cells as well as an efficient water surface correction procedure which guarantee the positivity of the computed fluid depth. We also design a new quadrature for the discretization of the source term, using which the new Scheme exactly preserves “lake at rest” steady states. We demonstrate these features of the new Scheme as well as its high resolution and robustness and its potential advantages over the triangular central-Upwind Scheme in a number of numerical examples.
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Well-balanced positivity preserving cell-vertex central-Upwind Scheme for shallow water flows
Computers & Fluids, 2016Co-Authors: Abdelaziz Beljadid, Abdolmajid Mohammadian, Alexander KurganovAbstract:Abstract We develop a new second-order two-dimensional central-Upwind Scheme on cell-vertex grids for approximating solutions of the Saint-Venant system with source terms due to bottom topography. Central-Upwind Schemes are developed based on the information about the local speeds of wave propagation. Compared to the triangular central-Upwind Schemes, the proposed cell-vertex one has an advantage of using more cell interfaces which provide more information on the waves propagating in different directions. We propose a new piecewise linear approximation of the bottom topography and a novel non-oscillatory reconstruction in which the gradient of each variable is computed using a modified minmod-type method to ensure the stability of the Scheme. A new technique is proposed for the correction of the water surface elevation which guarantees the positivity of the water depth. The well-balanced property of the proposed central-Upwind Scheme is ensured using a special discretization for the cell averages of the topography source terms. The proposed Scheme is tested on a number of numerical examples, among which we consider steady-state solutions with almost dry areas and their perturbations and solutions with rapidly varying flows over discontinuous bottom topography. Our numerical experiments confirm stability, well-balanced, positivity preserving properties and second-order accuracy of the proposed method. This Scheme can be applied to shallow water models when the bed topography is discontinuous and/or highly oscillatory, and on complicated domains where the use of unstructured grids is advantageous.
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well balanced central Upwind Scheme for a fully coupled shallow water system modeling flows over erodible bed
Journal of Computational Physics, 2015Co-Authors: Xin Liu, Alexander Kurganov, Abdolmajid Mohammadian, Julio Angel Infante SedanoAbstract:Intense sediment transport and rapid bed evolution are frequently observed under highly-energetic flows, and bed erosion sometimes is of the same magnitude as the flow itself. Simultaneous simulation of multiple physical processes requires a fully coupled system to achieve an accurate hydraulic and morphodynamical prediction. In this paper, we develop a high-order well-balanced finite-volume method for a new fully coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution.The nonequilibrium sediment transport equation is used to predict the sediment concentration variation. Since bed-load, sediment entrainment and deposition have significant effects on the bed evolution, an Exner-based equation is adopted together with the Grass bed-load formula and sediment entrainment and deposition models to calculate the morphological process. The resulting 5 × 5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-Upwind Scheme on a triangular grid. A computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. A special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed Scheme. The proposed fully coupled model is verified on a number of numerical experiments.
Frédéric Lagoutière - One of the best experts on this subject based on the ideXlab platform.
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Convergence order of Upwind type Schemes for transport equations with discontinuous coefficients
Journal de Mathématiques Pures et Appliquées, 2017Co-Authors: François Delarue, Frédéric Lagoutière, Nicolas VaucheletAbstract:An analysis of the error of the Upwind Scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the Upwind Scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other " diffusive " " first order " Schemes and to a forward semi-Lagrangian Scheme.
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probabilistic analysis of the Upwind Scheme for transport equations
Archive for Rational Mechanics and Analysis, 2011Co-Authors: François Delarue, Frédéric LagoutièreAbstract:We provide a probabilistic analysis of the Upwind Scheme for d-dimensional transport equations. We associate a Markov chain with the numerical Scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the Scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129–155, 2007) and Merlet (SIAM J Numer Anal 46(1):124–150, 2007): we prove that the Scheme is of order 1/2 in \({L^{\infty}([0,T],L^1(\mathbb R^d))}\) for an integrable initial datum of bounded variation and of order 1/2−e, for all e > 0, in \({L^{\infty}([0,T] \times \mathbb R^d)}\) for an initial datum of Lipschitz regularity. Our analysis provides a new interpretation of the numerical diffusion phenomenon.
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Probabilistic analysis of the Upwind Scheme for transport
Archive for Rational Mechanics and Analysis, 2007Co-Authors: François Delarue, Frédéric LagoutièreAbstract:We provide a probabilistic analysis of the Upwind Scheme for multi-dimensional transport equations. We associate a Markov chain with the numerical Scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the Scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we prove that the Scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon.
