A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

Zhang, Min; Huang, Weizhang; Qiu, Jianxian

doi:10.4208/cicp.oa-2021-0127

Cited by 12 publications

(5 citation statements)

References 46 publications

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“…In this section we describe a quasi-Lagrange MMDG method [27,38,41] for solving hyperbolic conservation laws in the form (1.1).…”

Section: The Moving Mesh Dg Methodsmentioning

confidence: 99%

“…Luo et al considered a quasi-Lagrange moving mesh DG method (MMDG) for conservation laws [27] and multi-component flows [28]. Zhang et al studied the MMDG solution for the radiative transfer equation [38,39] and shallow water equations (SWEs) [40,41]. Zhang et al [43] develop a arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the SWEs.…”

Section: Introductionmentioning

confidence: 99%

“…, u m ) T (m ≥ 1) is the unknown function, and the flux F(U, x) is an m-by-d matrix-valued function of U and x. A commonly used selection of time step in adaptive DG computations (e.g., see [27,40,41]) is…”

Section: Introductionmentioning

confidence: 99%

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A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

Zhang¹,

null²,

Qiu³

2023

NMTMA

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The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the α-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the α-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.

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“…In this section we describe a quasi-Lagrange MMDG method [27,38,41] for solving hyperbolic conservation laws in the form (1.1).…”

Section: The Moving Mesh Dg Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

Zhang¹,

null²,

Qiu³

2023

NMTMA

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“…Many numerical schemes have been proposed for solving these models, especially recently, high order schemes are very attractive due to their high resolutions for such problems with rich solution structures. For example, deterministic methods, there are finite difference, finite volume and finite element Eulerian methods [10, 12, 18-20, 43, 51, 53, 54,56,59], semi-Lagrangian methods [4,5,7,8,13,14,23,[31][32][33][34][35]38,45,46,52,58], and discontinuous Galerkin finite element methods [3,6,11,15,22,24,29,55,60], also see many other references therein. However, due to the highly oscillatory structure of such problems, linear type schemes for these problems would show significant spurious numerical oscillations, which might get worse with increased orders.…”

Section: Introductionmentioning

confidence: 99%

A High Order Bound Preserving Finite Difference Linear Scheme for Incompressible Flows

Zhang¹,

Xiong²

2022

CiCP

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We propose a high order finite difference linear scheme combined with a high order bound preserving maximum-principle-preserving (MPP) flux limiter to solve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much less costly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP flux limiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guidingcenter model and the incompressible Euler system. The comparison between the linear and WENO type schemes, with and without the MPP flux limiter, will demonstrate the good performance of our proposed approach.

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“…Caleffi [3] developed a well-balanced fourth-order finite volume Hermite WENO scheme for the one-dimensional SWEs on the basis of [19]. For more related well-balanced high-order methods, e.g., finite difference schemes [6,7,8,14,16,25], finite volume schemes [5,11,18,28], and DG methods [12,30,31,33,34,35].…”

Section: Introductionmentioning

confidence: 99%

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

Zhao¹,

Zhang²

2022

Preprint

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In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balance property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput. Phys., 208 (2005), 206-227.] to balance the flux gradients and the source terms. The fluxes in the original equation are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balance property. Instead of using a five-point stencil in the same fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction.Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.

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A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

Cited by 12 publications

References 46 publications

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

A High Order Bound Preserving Finite Difference Linear Scheme for Incompressible Flows

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

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