High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

Christlieb, Andrew; Liu, Yuan; Tang, Qi; Xu, Zhengfu

doi:10.1016/j.jcp.2014.10.029

Cited by 46 publications

(29 citation statements)

References 29 publications

(44 reference statements)

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“…The main idea of WENO schemes is a non-linear-weighted combination of several local reconstructions based on different stencils and the usage of it as a final WENO reconstruction. Now, many improved WENO scheme are developed, such as Hermite-WENO [31], hybrid-WENO [32], positivity-preserving WENO [33], ADER-WENO [34], CWENO [35] and so on. The finite difference and the finite volume WENO schemes are widely used in many areas [36][37][38][39][40][41].…”

Section: Literature Surveymentioning

confidence: 99%

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

Dong

Yang

2018

Mathematics

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We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

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Section: Literature Surveymentioning

confidence: 99%

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

Dong

Yang

2018

Mathematics

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“…Zhang et al constructed a genuinely high-order maximum-principle-satisfying finite volume schemes for multi-dimensional nonlinear scalar conservation laws on both rectangular meshes [33] and triangular meshes [34] by limiting the reconstructed polynomials around cell averages. The flux limiting technique developed by Christlieb et al [35] is another family of maximum-principle-satisfying methods on unstructured meshes. In our scheme, it is proposed that the extrema of numerical solutions are measured by extrema of polynomial on a cluster of points, following the technique of Zhang et al [33,34].…”

Section: Introductionmentioning

confidence: 99%

An Integrated Quadratic Reconstruction for Finite Volume Schemes to Scalar Conservation Laws in Multiple Dimensions

Chen¹

2020

CSIAM-AM

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We proposed a piecewise quadratic reconstruction method in multiple dimensions, which is in an integrated style, for finite volume schemes to scalar conservation laws. This integrated quadratic reconstruction is parameter-free and applicable on flexible grids. We show that the finite volume schemes with the new reconstruction satisfy a local maximum principle with properly setup on time steplength. Numerical examples are presented to show that the proposed scheme attains a third-order accuracy for smooth solutions in both 2D and 3D cases. It is indicated by numerical results that the local maximum principle is helpful to prevent overshoots in numerical solutions.

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“…To reduce inherent numerical dissipation of these first-order schemes, high-order reconstruction of variables can be used. This can be achieved using either monotone upstream-centered schemes for conservation laws (MUSCL) methods (Van Leer, 1974; Sweby, 1984; Mallison et al , 2005), which use total variation diminishing (TVD) concept or weighted/essentially non-oscillatory (W/ENO) (Liu and Osher, 1998; Qiu and Shu, 2002; Christlieb et al , 2015) methods. In this work, both MUSCL and weighted essentially non-oscillatory (WENO) reconstructions are used for simulation of the system of conservation laws.…”

Section: Introductionmentioning

confidence: 99%

A comparative study of explicit high-resolution schemes for compositional simulations

Moshiri

Manzari

2019

HFF

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Purpose This paper aims to numerically study the compositional flow of two- and three-phase fluids in one-dimensional porous media and to make a comparison between several upwind and central numerical schemes. Design/methodology/approach Implicit pressure explicit composition (IMPEC) procedure is used for discretization of governing equations. The pressure equation is solved implicitly, whereas the mass conservation equations are solved explicitly using different upwind (UPW) and central (CEN) numerical schemes. These include classical upwind (UPW-CLS), flux-based decomposition upwind (UPW-FLX), variable-based decomposition upwind (UPW-VAR), Roe’s upwind (UPW-ROE), local Lax–Friedrichs (CEN-LLF), dominant wave (CEN-DW), Harten–Lax–van Leer (HLL) and newly proposed modified dominant wave (CEN-MDW) schemes. To achieve higher resolution, high-order data generated by either monotone upstream-centered schemes for conservation laws (MUSCL) or weighted essentially non-oscillatory (WENO) reconstructions are used. Findings It was found that the new CEN-MDW scheme can accurately solve multiphase compositional flow equations. This scheme uses most of the information in flux function while it has a moderate computational cost as a consequence of using simple algebraic formula for the wave speed approximation. Moreover, numerically calculated wave structure is shown to be used as a tool for a priori estimation of problematic regions, i.e. degenerate, umbilic and elliptic points, which require applying correction procedures to produce physically acceptable (entropy) solutions. Research limitations/implications This paper is concerned with one-dimensional study of compositional two- and three-phase flows in porous media. Temperature is assumed constant and the physical model accounts for miscibility and compressibility of fluids, whereas gravity and capillary effects are neglected. Practical implications The proposed numerical scheme can be efficiently used for solving two- and three-phase compositional flows in porous media with a low computational cost which is especially useful when the number of chemical species increases. Originality/value A new central scheme is proposed that leads to improved accuracy and computational efficiency. Moreover, to the best of authors knowledge, this is the first time that the wave structure of compositional model is investigated numerically to determine the problematic situations during numerical solution and adopt appropriate correction techniques.

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High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

Cited by 46 publications

References 29 publications

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

An Integrated Quadratic Reconstruction for Finite Volume Schemes to Scalar Conservation Laws in Multiple Dimensions

A comparative study of explicit high-resolution schemes for compositional simulations

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