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Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

Jun Zhu, Chi-Wang Shu
2019 Communications on Applied Mathematics and Computation  
In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine  ...  This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state problems  ...  [26, 29] proposed a combination of a finite-volume discretization method with dissipative terms of third-order and a Runge-Kutta time stepping method to yield an effective method for solving the Euler  ... 
doi:10.1007/s42967-019-00044-7 fatcat:mghm7qljkrclndaaqd53pwnxty

High-order Finite Difference and Finite Volume WENO Schemes and Discontinuous Galerkin Methods for CFD

Chi-Wang Shu
2003 International journal of computational fluid dynamics (Print)  
In this paper we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory WENO nite di erence methods, the WENO nite volume methods, and the  ...  Key words. weighted essentially non-oscillatory, discontinuous Galerkin, nite di erence method, nite volume method, computational uid dynamics Subject classi cation.  ...  HIGH ORDER FINITE DIFFERENCE AND FINITE VOLUME WENO SCHEMES AND DISCONTINUOUS GALERKIN METHODS FOR CFD CHI-WANG SHU Abstract.  ... 
doi:10.1080/1061856031000104851 fatcat:ysiswr2x4nal3nuinmuq5wvui4

High-order finite volume WENO schemes for the shallow water equations with dry states

Yulong Xing, Chi-Wang Shu
2011 Advances in Water Resources  
A high-order accurate finite volume weighted essentially non-oscillatory (WENO) scheme is proposed in this paper to address these difficulties and to provide an efficient and robust method for solving  ...  It is desirable to develop numerical methods which preserve exactly these steady state solutions.  ...  For the finite volume WENO schemes, an extra Hermite type reconstruction was introduced in [40] to compute them.  ... 
doi:10.1016/j.advwatres.2011.05.008 fatcat:7n7c3vp25rbfpdyxjqw6bu3zoi

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

Changna Lu, Luoyan Xie, Hongwei Yang
2018 Mathematical Problems in Engineering  
These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method  ...  A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow  ...  For 2D finite volume case, we cannot simply proceed in a dimension-by-dimension fashion as in finite difference method.  ... 
doi:10.1155/2018/2652367 fatcat:de7bnpd2qngqzdfwrwt3gi7qou

Weighted Essentially Non-oscillatory Schemes on Triangular Meshes

Changqing Hu, Chi-Wang Shu
1999 Journal of Computational Physics  
Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations.  ...  In this paper we construct high-order weighted essentially non-oscillatory schemes on two-dimensional unstructured meshes (triangles) in the finite volume formulation.  ...  Djomehri on using adaptive methods with WENO schemes, that this grouping may fail near adaptively refined regions where triangle sizes change very abruptly.  ... 
doi:10.1006/jcph.1998.6165 fatcat:rafdse47wzac7nhhjqrfr7ustu

A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids

Hong Luo, Joseph D. Baum, Rainald Löhner
2007 Journal of Computational Physics  
A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for a discontinuous Galerkin finite element method on unstructured grids.  ...  The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method.  ...  Cockburn at University of Minnesota for many helpful, instructive, and fruitful discussions about DG and WENO.  ... 
doi:10.1016/j.jcp.2006.12.017 fatcat:su5p2w575zggtdyaplkzddq2ia

A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids

Guanghui Hu, Ruo Li, Tao Tang
2010 Journal of Computational Physics  
A robust high-order algorithm is proposed to solve steady Euler equations on unstructured grids.  ...  Shu, Hierarchical reconstruction for discontinuous galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, Journal of Computational Physics 228 (2009  ...  Then the polynomial is used directly to describe the steady-state of Euler equations in an integrated form.  ... 
doi:10.1016/j.jcp.2009.11.002 fatcat:5ekw2ndgtndhxifhyztiqngnem

A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids

Hong Luo, Yidong Xia, Shujie Li, Robert Nourgaliev, Chunpei Cai
2012 Journal of Computational Physics  
method, is presented for solving the compressible Euler equations on tetrahedral grids.  ...  A Hermite WENO reconstruction-based discontinuous Galerkin method RDG(P 1 P 2 ), designed not only to enhance the accuracy of discontinuous Galerkin method but also to ensure linear stability of the RDG  ...  Indeed, compared to the finite element methods and finite volume methods, the DGM require solutions of systems of equations with more unknowns for the same grids.  ... 
doi:10.1016/j.jcp.2012.05.011 fatcat:klv6c72ajzc43ee7bnygskl5p4

Conservative finite volume scheme for first-order viscous relativistic hydrodynamics [article]

Alex Pandya, Elias R. Most, Frans Pretorius
2022 arXiv   pre-print
We present the first conservative finite volume numerical scheme for the causal, stable relativistic Navier-Stokes equations developed by Bemfica, Disconzi, Noronha, and Kovtun (BDNK).  ...  We assess the robustness of our new method in a series of flat-spacetime tests for a conformal fluid, and provide a detailed comparison with previous approaches of Pandya & Pretorius (2021).  ...  III D outlines the reconstruction of primitive variables, for which we use the WENO method [54] .  ... 
arXiv:2201.12317v2 fatcat:ziczz4stpjdfpcnqknn6k7hdu4

WENO schemes on unstructured meshes using a relaxed a posteriori MOOD limiting approach

Pericles S. Farmakis, Panagiotis Tsoutsanis, Xesús Nogueira
2020 Computer Methods in Applied Mechanics and Engineering  
The augmented numerical schemes are applied to the 2D unsteady Euler equations for a multitude of test problems including the 2D vortex evolution, cylindrical explosion, double-Mach reflection, and an  ...  In this paper a relaxed formulation of the a posteriori Multi-dimensional Optimal Order Detection (MOOD) limiting approach is introduced for weighted essentially non-oscillatory (WENO) finite volume schemes  ...  equations [73] , since it has an intrinsic correspondence with an instability mechanism of the Euler equations [74, 75] .  ... 
doi:10.1016/j.cma.2020.112921 fatcat:2ecbh4wk4jau5hmjsaceawy2eq

A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes

Yunrui Guo
2018 Advances in Applied Mathematics and Mechanics  
We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO (SWENO) scheme.  ...  However, weak oscillations still appear when the SWENO scheme deals with complex low density equations.  ...  Table 4 : 4 Accuracy for 2D Burgers equation with scaling limiter. Table 5 : 5 Accuracy for 2D Burgers equation without scaling limiter.  ... 
doi:10.4208/aamm.oa-2016-0196 fatcat:mzenatpzszfpfd3ckuxuvdxjzm

A class of new high-order finite-volume TENO schemes for hyperbolic conservation laws with unstructured meshes [article]

Zhe Ji and Tian Liang and Lin Fu
2022 arXiv   pre-print
For conceptual verification, the TENO schemes with third- to sixth-order accuracy are constructed.  ...  In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed.  ...  Fundamentals of the unstructured finite-volume methods In this section, the basic concepts of the unstructured finite-volume method including the classical high-order WENO and CWENO reconstruction schemes  ... 
arXiv:2105.02127v2 fatcat:yqxwn4fi3zckxaz2ujrnce5iym

Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws [chapter]

Chi-Wang Shu
1998 Lecture notes in mathematics  
ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities.  ...  The reconstruction in [38] is a natural extension of an earlier second order version of Harten and Osher [37] .  ...  Finite volume 2D scalar ENO and WENO. 1.  ... 
doi:10.1007/bfb0096355 fatcat:dd3cyuuouzgmfjitiphcu2uuv4

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations [article]

Mirco Ciallella, Lorenzo Micalizzi, Philipp Öffner, Davide Torlo
2021 arXiv   pre-print
In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow  ...  water equations.  ...  All authors would like to thank Jonatan Núñez for sharing his high-order FV-WENO code on his repository [45] . We have started our work by adapting his code.  ... 
arXiv:2110.13509v1 fatcat:dfpbz67dnvgpdpn3qd73gqdjke

Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case

Jianxian Qiu, Chi-Wang Shu
2004 Journal of Computational Physics  
For this reason, the HWENO finite volume methodology is more suitable to serve as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods, than the original WENO finite volume methodology.  ...  The construction of HWENO schemes is based on a finite volume formulation, Hermite interpolation, and nonlinearly stable Runge-Kutta methods.  ...  finite volume WENO schemes in [26] and DG3 with TVB limiter [7] .  ... 
doi:10.1016/j.jcp.2003.07.026 fatcat:g6heetd4qngktine77xjauk2j4
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