P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

Feng, Xinlong; Li, Rongfei; He, Ya‐Ling; Liu, Demin

doi:10.1007/s10915-011-9557-4

Cited by 24 publications

(15 citation statements)

References 36 publications

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“…. (27) It should be noted that for the lake at rest with , we have and , and the designed quadratures (26) and (27) satisfy (23).…”

Section: Well-balanced Discretization Of the Source Termmentioning

confidence: 99%

“…Consider the proposed central-upwind scheme (8), (9), (22), (26) and (27) for the system (3)- (7). Assume that the forward Euler method is used for solving the system of ODEs (8) while for all at time .…”

Section: Theoremmentioning

confidence: 99%

“…In order to show the importance of well-balanced property of the scheme, we apply a non-wellbalanced version of our scheme to the initial-boundary value problem with . To obtain the non-well-balanced scheme, the proposed well-balanced quadratures (26) and (27) are replaced with a straightforward midpoint rule discretization:…”

Section: Example 2 -Small Perturbation Over An Exponential Humpmentioning

confidence: 99%

“…In this paper we consider the two-dimensional (2D) shallow water equations (SWEs): (1) The main goal of this paper is to develop a second-order well-balanced positivity preserving central-upwind scheme for (1) on unstructured quadrilateral grids. Such grids have been widely used in finite volume methods for various applications, in particular, for numerically solving incompressible Navier-Stokes, diffusion equations, semilinear elliptic and elliptic systems, see, e.g., [25][26][27][28] and references therein. In particular, quadrilateral grids have been used to develop finite volume methods for the 2D SWEs, see, e.g., [29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

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A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

et al. 2016

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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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“…. (27) It should be noted that for the lake at rest with , we have and , and the designed quadratures (26) and (27) satisfy (23).…”

Section: Well-balanced Discretization Of the Source Termmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Example 2 -Small Perturbation Over An Exponential Humpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

et al. 2016

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“…Note that we can also use the interpolation finite element method to deal with the nonlinear term ν(θ h ), κ(θ h ) and β(θ h ) respectively. For further details, readers can refer to [18].…”

Section: Mixed Finite Element Methodsmentioning

confidence: 99%

Implicit–explicit schemes of finite element method for the non-stationary thermal convection problems with temperature-dependent coefficients

Zhang

Feng

Yuan

2016

International Communications in Heat and Mass Transfer

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For the time-dependent thermal convection problems with temperature-dependent coefficients, the implicit-explicit scheme is presented, in which mixed finite element method is applied for the spatial approximation of the velocity, pressure and temperature while the time discretization is based on the high-order backward difference scheme. Linear terms are dealt with the implicit scheme while the nonlinear terms are treated by the semi-implicit scheme. The advantages for this scheme are unconditionally stable, decoupled computational and second order accuracy. Finally, numerical tests illustrate the theoretical results of the presented schemes, and display that the highly efficient method conserves the property of divergence free of the original problems.

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A vertex‐centered linearity‐preserving discretization of diffusion problems on polygonal meshes

Gao

Dai

2015

Numerical Methods in Fluids

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Summary This paper introduces a vertex‐centered linearity‐preserving finite volume scheme for the heterogeneous anisotropic diffusion equations on general polygonal meshes. The unknowns of this scheme are purely the values at the mesh vertices, and no auxiliary unknowns are utilized. The scheme is locally conservative with respect to the dual mesh, captures exactly the linear solutions, leads to a symmetric positive definite matrix, and yields a nine‐point stencil on structured quadrilateral meshes. The coercivity of the scheme is rigorously analyzed on arbitrary mesh size under some weak geometry assumptions. Also, the relation with the finite volume element method is discussed. Finally, some numerical tests show the optimal convergence rates for the discrete solution and flux on various mesh types and for various diffusion tensors. Copyright © 2015 John Wiley & Sons, Ltd.

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P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

Cited by 24 publications

References 36 publications

A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

Implicit–explicit schemes of finite element method for the non-stationary thermal convection problems with temperature-dependent coefficients

A vertex‐centered linearity‐preserving discretization of diffusion problems on polygonal meshes

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