Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-02T18:05:22.587Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes

Published online by Cambridge University Press:  20 August 2015

Rui Zhang ,
Mengping Zhang  and
Chi-Wang Shu
Rui Zhang*
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Mengping Zhang*
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.

Keywords

65M08Weighted essentially non-oscillatory (WENO) schemesfinite volume schemesaccuracy
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 807 - 827
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]Balsara, D. S. and Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160 (2000), 405–452.Google Scholar
[2]Borges, R., Carmona, M., Costa, B. and Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), 3191–3211.Google Scholar
[3]Casper, J., Shu, C.-W. and Atkins, H. L., Comparison of two formulations for high-order accurate essentially nonoscillatory schemes, AIAA J., 32 (1994), 1970–1977.Google Scholar
[4]Gottlieb, S., Ketcheson, D. I. and Shu, C.-W., High order strong stability preserving time discretizations, J. Sci. Comput., 38 (2009), 251–289.Google Scholar
[5]Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357–393.Google Scholar
[6]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), 231–303.CrossRefGoogle Scholar
[7]Henrick, A. K., Aslam, T. D. and Powers, J. M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207 (2005), 542–567.CrossRefGoogle Scholar
[8]Jiang, G.-S. and Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.Google Scholar
[9]Liu, X.-D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200–212.CrossRefGoogle Scholar
[10]Liu, Y.-Y., Shu, C.-W. and Zhang, M., On the positivity of linear weights in WENO approximations, Acta. Math. Appl. Sinica., 25 (2009), 503–538.CrossRefGoogle Scholar
[11]Qiu, J., Khoo, B. C. and Shu, C.-W., A numerical study for the performanceof the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes, J. Comput. Phys., 212 (2006), 540–565.Google Scholar
[12]Serna, S. and Marquina, A., Power ENO methods: a fifth-order accurate weighted power ENO method, J. Comput. Phys., 194 (2004), 632–658.Google Scholar
[13]Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Cockburn, B., Johnson, C., Shu, C.-W. and Tadmor, E. (Editor: Quarteroni, A.), Lecture Notes in Mathematics, Volume 1697, Springer, Berlin, 1998, 325–432.Google Scholar
[14]Shu, C.-W., High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), 82–126.CrossRefGoogle Scholar
[15]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439–471.Google Scholar
[16]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1997.Google Scholar
[17]Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115–173.Google Scholar
[18]Zhang, W. and MacFadyen, A., RAM: a relativistic adaptive mesh refinement hydrodynamics code, Astrophys. J. Supp. Ser., 164 (2006), 255–279.Google Scholar