Skip to main content Accessibility help
×
Home
Hostname: page-component-544b6db54f-dkqnh Total loading time: 0.255 Render date: 2021-10-24T07:33:55.908Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids

Published online by Cambridge University Press:  20 August 2015

Guanghui Hu*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Ruo Li*
Affiliation:
CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China
Tao Tang*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
Corresponding author.Email:ghhu@math.msu.edu
Get access

Abstract

A recent work of Li et al. [Numer. Math. Theor. Meth. Appl., 1(2008), pp. 92-112] proposed a finite volume solver to solve 2D steady Euler equations. Although the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the shock region, the overshoot or undershoot phenomenon can still be observed. Moreover, the numerical accuracy is degraded by using Venkatakrishnan limiter. To fix the problems, in this paper the WENO type reconstruction is employed to gain both the accurate approximations in smooth region and non-oscillatory sharp profiles near the shock discontinuity. The numerical experiments will demonstrate the efficiency and robustness of the proposed numerical strategy.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barth, T. J., Recent developments in high order k-exact reconstruction on unstructured meshes, AIAA Paper, 93 (993), 0668.Google Scholar
[2]Barth, T. J. and Deconinck, H., High-Order methods for Computational Physics, Springer, 1999.CrossRefGoogle Scholar
[3]Barth, T. J. and Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, AIAA Paper, 89 (1989), 0366.Google Scholar
[4]Bassi, F. and Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138 (1997), 251285.CrossRefGoogle Scholar
[5]Batten, P., Leschziner, M. A. and Goldberg, U. C., Average-state Jacobians and implicit methods for compressible viscous and turbulent flows, J. Comput. Phys., 137 (1997), 3878.CrossRefGoogle Scholar
[6]Blazek, J., Computational Fluid Dynamics: Principles and Applications, Elsevier Science Publication, 2005.Google Scholar
[7]Chen, R. F. and Wang, Z. J., Fast block lower-upper symmetric Gauss-Seidel scheme for arbitrary grid, AIAA J., 38(12) (2000), 22382245.CrossRefGoogle Scholar
[8]Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357393.CrossRefGoogle Scholar
[9]Harten, A., On a class of high resolution total variation stable finite difference schemes, SIAM J. Numer. Anal., 21 (1984), 123.CrossRefGoogle Scholar
[10]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), 231303.CrossRefGoogle Scholar
[11]Harten, A., Engquist, B., Osher, S. and Chakravathy, R., Some results on uniformly high order accurate essentially non-oscillatory schemes, Appl. Numer. Math., 2 (1986), 347377.CrossRefGoogle Scholar
[12]Hu, C. Q. and Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), 97127.CrossRefGoogle Scholar
[13]Hu, G. H., Li, R. and Tang, T., A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids, J. Comput. Phys., 229 (2010), 16811697.CrossRefGoogle Scholar
[14]Jiang, G. and Shu, C. W., Efficient implementation of weighted ENO, J. Comput. Phys., 126 (1996), 202228.CrossRefGoogle Scholar
[15]Leveque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.CrossRefGoogle Scholar
[17]Li, R., Wang, X. and Zhao, W. B., A multigrid block lower-upper symmetric Gauss-Seidel algorithm for steady Euler equation on unstructured grids, Numer. Math. Theor. Meth. Appl., 1 (2008), 92112.Google Scholar
[18]Liu, X. D. and Osher, S., Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids, J. Comput. Phys., 142 (1998), 304338.CrossRefGoogle Scholar
[19]Liu, Y. J., Shu, C. W., Tadmor, E. and Zhang, M. P., Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal., 45 (2007), 24422467.CrossRefGoogle Scholar
[20]Liu, Y. J., Shu, C. W., Tadmor, E. and Zhang, M. P., Non-oscillatory hierarchical reconstruction for central and finite volume schemes, Commun. Comput. Phys., 2(5) (2007), 933963.Google Scholar
[21]Luo, H., Baum, J. D. and Lohner, R., A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids, J. Comput. Phys., 225 (2007), 686713.CrossRefGoogle Scholar
[22]Luo, H., Baum, J. D. and Lohner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227 (2008), 88758893.CrossRefGoogle Scholar
[23]Michalak, K. and Ollivier-Gooch, C., Limiters for unstructured higher-order accurate solutions of the Euler equations, AIAA Forty-Sixty Aerospace Sciences Meeting, 2008.Google Scholar
[24]Mitchell, C. R. and Walters, R. W., K-exact reconstruction for the Navier-Stokes equation on arbitrary grids, AIAA Paper, 93 (1993), 0536.Google Scholar
[26]Qiu, J. X. and Shu, C. W., Hermite WENO schemes and their application aslimiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case, J. Comput. Phys., 193 (2004), 115135.CrossRefGoogle Scholar
[27]Shu, C. W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, Springer Berlin, pp. 325432, 1998.Google Scholar
[28]Titarev, V. A., Tsoutsanis, P. and Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8 (2010), 585609.Google Scholar
[29]Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118 (1995), 120130.CrossRefGoogle Scholar
[30]Xu, Z. L., Liu, Y. J. and Shu, C. W., Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, J. Comput. Phys., 228 (2009), 21942212.CrossRefGoogle Scholar
[31]Zhang, Y. T. and Shu, C. W., Third order WENO scheme on three dimensional tetrahedral meshes, Commun. Comput. Phys., 5 (2009), 836848.Google Scholar
[32]Zhu, J. and Qiu, J. X., Trigonometric WENO schemes for hyperbolic conservation laws and highly oscillatory problems, Commun. Comput. Phys., 8 (2010), 12421263.Google Scholar
[33]Zhu, J., Qiu, J. X., Shu, C. W. and Dumbser, M., Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes, J. Comput. Phys., 227 (2008), 43304353.CrossRefGoogle Scholar
33
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *