Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids

Lou, Jialin; Liu, Xiaodong; Luo, Hong; Nishikawa, Hiroaki

doi:10.4208/cicp.oa-2017-0186

Cited by 15 publications

(30 citation statements)

References 33 publications

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“…The extension to curvilinear coordinates using WCNS [51] and field aligned unstructured meshes by discontinuous galerkin methods [23,52] is currently ongoing and the results will be discussed elsewhere.…”

Section: Discussionmentioning

confidence: 99%

“…This radical approach has been shown to offer several advantages over conventional methods, such as accelerated convergence for steady state solution and higher order of accuracy for both primary and gradient variables, as demonstrated for diffusion [16], the incompressible/compressible Navier-Stokes equations [17,18], third-order dispersion equations [19], an incompressible magnetohydrodynamics model [20], an elliptic distance-function model [21], and so on. The original approach of Nishikawa has been further extended to a constant diffusion tensor by Lou et al [22,23] discretized with the reconstructed discontinuous Galerkin scheme (rDG). The hyperbolic approach has also been implemented for an anisotropic diffusion equation, based on highorder finite-volume schemes in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…where L r = 1/(2π) is a relaxation length. In the hyperbolic approach proposed by Lou et al [22], based on reconstructed Discontinuous Galerkin scheme (rDG), ν opt is taken as 1.…”

mentioning

confidence: 99%

“…3a For the first test case, the exact solution is given byT (x, y) = 1 − tanh (x−0.5) 2 +(y−0.5) 20.01 and the diffusion tensor given by Lou et al[22,23] is modified by including extreme anisotropy through D || ,…”

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confidence: 99%

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First order hyperbolic approach for Anisotropic Diffusion equation

Chamarthi

Nishikawa

Komurasaki

2019

Journal of Computational Physics

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In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in Refs. [J. Comput. Phys., 284 (2015) 59-69 and 374 (2018) 1120-1151]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…where L r = 1/(2π) is a relaxation length. In the hyperbolic approach proposed by Lou et al [22], based on reconstructed Discontinuous Galerkin scheme (rDG), ν opt is taken as 1.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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First order hyperbolic approach for Anisotropic Diffusion equation

Chamarthi

Nishikawa

Komurasaki

2019

Journal of Computational Physics

Self Cite

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“…2, uniform structured grids, nonuniform unstructured grids together with a specially designed heterogeneous grids are used in this test case and grid convergence studies based on the three different types of grids are performed. It should be pointed out that the heterogeneous grids are specially designed, inspired by the work [31], as this type grids can be really effective in testing the performance of the viscous discretization under the demanding situations.…”

Section: Numerical Examples 41 a Model Problem With A Source Termmentioning

confidence: 99%

A Direct Discontinuous Galerkin Method with Interface Correction for the Compressible Navier-Stokes Equations on Unstructured Grids

Cheng¹

2018

AAMM

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Since the original DDG method has been introduced by Liu et al. [8] in 2009, a variety of DDG type methods have been proposed and further developed. In this paper, we further investigate and develop a new DDG method with interface correction (DDG (IC)) as the discretization of viscous and heat fluxes for the compressible Navier-Stokes equations on unstructured grids. Compared to the original DDG method, the newly developed DDG (IC) method demonstrates its superior in delivering the optimal order of accuracy under demanding situations. Strategies in extension and application of this newly developed DDG (IC) method for solving the compressible Navier-Stokes equations and special treatments designed for handling boundary viscous fluxes are presented and examined in this work. The performance of the new DDG method with interface correction is carefully evaluated and assessed through a number of typical test cases. Numerical experiments show that the new DDG method with interface correction can achieve the optimal order of accuracy on both uniform structured grids and nonuniform unstructured grids, which clearly indicates its potential for further applications of real engineering practices.

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