A Fifth Order Flux Implicit WENO Method

Gottlieb, Sigal; Mullen, Julia; Ruuth, Steven J.

doi:10.1007/s10915-005-9034-z

Cited by 36 publications

(19 citation statements)

References 8 publications

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“…However, in most papers about WENO schemes only the explicit methods are applied. There are some works considering implicit finite-difference WENO schemes [11], but the obtained results were not very encouraging. The main reason why the implicit RungeKutta methods were not usually used in combination with the WENO schemes was based on their bad stability properties and high computational cost.…”

Section: Temporal Discretizationmentioning

confidence: 88%

“…In [5] well-balanced semi-implicit first order numerical schemes for the open-channel flow equations were developed, based on the balancing property of the explicit numerical schemes presented in [29]. The idea of constructing the high order implicit finite-difference WENO schemes was considered in [11], but the stability properties for the constructed schemes were satisfied just for quite limited time steps.…”

Section: Introductionmentioning

confidence: 99%

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High order accurate semi-implicit WENO schemes for hyperbolic balance laws

Črnjarić-Žic

Crnković

2011

Applied Mathematics and Computation

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Section: Temporal Discretizationmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

Črnjarić-Žic

Crnković

2011

Applied Mathematics and Computation

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“…The discretization of the first term in the right-hand side of Equation (23) can be achieved in a variety of ways. In this paper, we consider the Lax-Friedrichs flux splitting, which uses three stencils, formed by five points [58]. In this method, the convection term is calculated by flux terms, which are determined by:…”

Section: Solving the Cahn-hillard Equation For Interface Capturing Wimentioning

confidence: 99%

A Comparative Study of Multiphase Lattice Boltzmann Methods for Bubble-Dendrite Interaction during Solidification of Alloys

2018

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This paper presents a comparative study between the pseudopotential Shan-Chen model and the phase field multiphase lattice Boltzmann method for simulating bubble dynamics during dendritic solidification of binary alloys. The Shan-Chen method is an efficient lattice Boltzmann multiphase method despite having some limitations, including the generation of large spurious currents. The phase field model solves the Cahn-Hilliard equation in addition to the Navier-Stokes equation to track the interface between phases. The phase field method is more accurate than the Shan-Chen model for simulation of fluids with a high-density ratio since it generates an acceptable small spurious current, though at the expense of higher computational costs. For the simulations in this article, the multiphase lattice Boltzmann model was coupled with the cellular automata and finite difference methods to solve temperature and concentration fields. The simulated results were presented and compared regarding the ability of each model to simulate phenomena at a microscale resolution, such as Marangoni convection, the magnitude of spurious current, and the computational costs. It is shown that although Shan-Chen methods can replicate some qualitative features of bubble-dendrite interaction, the generated spurious current is unacceptably large, particularly for practical values of the density ratio between fluid and gas phases. This occurs even after implementation of several enhancements to the original Shan-Chen method. This serious limitation makes the Shan-Chen models unsuitable to simulate fluid flow phenomena, such as Marangoni convection, because the large spurious currents mask completely the physical flow.

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“…There have been attempts to combine implicit time integrators with WENO methods on uniform meshes. A fifth order flux implicit WENO method on uniform meshes is studied in [10].…”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of hyperbolic equations with singular source terms

Turk¹,

Ashyraliyev²

2014

AIP Conference Proceedings

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Articles you may be interested inOn the numerical solution of diffusion problem with singular source terms AIP Conf.Abstract. A numerical study for hyperbolic equations having singular source terms is presented. Singular in the sense that within the spatial domain the source is defined by a Dirac delta function. Solutions of such problems will have discontinuities which forms an obstacle for standard numerical methods. In this paper, a fifth order flux implicit WENO method with non-uniform meshes is studied for approximate solutions of hyperbolic equations having singular source terms. Numerical examples are provided.

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A Fifth Order Flux Implicit WENO Method

Cited by 36 publications

References 8 publications

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

A Comparative Study of Multiphase Lattice Boltzmann Methods for Bubble-Dendrite Interaction during Solidification of Alloys

On the numerical solution of hyperbolic equations with singular source terms

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