Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T00:00:01.588Z Has data issue: false hasContentIssue false
  • English
  • Français

A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

Part of: Shock waves and blast waves Partial differential equations, initial value and time-dependent initial-boundary value problems Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  15 October 2015

Ziyao Sun ,
Honghui Teng  and
Feng Xiao
Ziyao Sun
Affiliation:
Department of Energy Sciences, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama, 226-8502, Japan
Honghui Teng
Affiliation:
State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China
Feng Xiao*
Affiliation:
Department of Energy Sciences, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama, 226-8502, Japan
*
*Corresponding author. Email addresses: [email protected] (Z. Sun), [email protected] (H. Teng), [email protected] (F. Xiao)
Get access

Abstract

This paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

Keywords

Multi-moment finite volume methodWENOflux reconstructioncompressible flowconservative methodoscillation-suppressing

MSC classification

Secondary: 65M08: Finite volume methods 65M70: Spectral, collocation and related methods 76L05: Shock waves and blast waves 76N15: Gas dynamics, general
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 4 , October 2015 , pp. 901 - 930
Copyright
Copyright © Global-Science Press 2015 

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]Borges, R., Carmona, M., Costa, B. and Don, W.S., An improved weighted essentially nonoscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), 3191–3211.CrossRefGoogle Scholar
[2]Chen, C.G., Li, X.L., Shen, X.S. and Xiao, F., Global shallow water models based on multi-moment constrained finite volume method and three quasi-uniform spherical grids, J. Comp. Phys. 271(2014), 191–223.Google Scholar
[3]Chen, C.G. and Xiao, F., Shallow water model on cubed-sphere by multi-moment finite volume method, J. Comput. Phys. 227(2008), 5019–5044.CrossRefGoogle Scholar
[4]Cockburn, B. and Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comput., 52 (1989), 411–435.Google Scholar
[5]Cockburn, B., Lin, S.Y. and Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys., 84 (1989), 90–113.Google Scholar
[6]Cockburn, B., Hou, S. and Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comput., 54 (1990), 545–581.Google Scholar
[7]Du, J., Shu, C.W. and Zhang, M.P., A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework, Applied Numerical Mathematics, to appear (2014).Google Scholar
[8]Henrick, A.K., Aslam, T.D. and Powers, J.M., Mapped weighted essentially nonoscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207 (2005), 542–567.CrossRefGoogle Scholar
[9]Hesthaven, J.S. and Warburton, T., Nodal discontinuous Galerkin methods: Algorithms, analysis and applications, Springer, 2008.Google Scholar
[10]Hu, C.Q. and Shu, C.W., Weighted Essentially Non-oscillatory Schemes on Triangular Meshes, J. Comput. Phys., 150 (1999), 97–127.Google Scholar
[11]Huang, C.S., Xiao, F. and Arbogast, T., Fifth Order Multi-moment WENO Schemes for Hyperbolic Conservation Laws, Journal of Scientific Computing, in press (2014), DOI 10.1007/s10915-014-9940-z.Google Scholar
[12]Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA Paper, 2007-4079 (2007).Google Scholar
[13]Ii, S. and Xiao, F., CIP/multi-moment finite volume method for Euler equations, a semi-Lagrangian characteristic formulation, J. Comput. Phys., 222(2007), 849–871.CrossRefGoogle Scholar
[14]Ii, S. and Xiao, F., High order multi-moment constrained finite volume method. Part I: Basic formulation, J. Comput. Phys., 228(2009), 3669–3707.Google Scholar
[15]Jiang, G.S. and Shu, C.W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.CrossRefGoogle Scholar
[16]Kopriva, D.A. and Kolias, J.H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comput. Phys., 125(1996), 244–261.Google Scholar
[17]Li, X.L., Chen, C.G., Shen, X.S. and Xiao, F., A Multi-moment Constrained Finite-Volume Model for Nonhydrostatic Atmospheric Dynamics, Mon. Wea. Rev., 141(2013), 1216–1240.Google Scholar
[18]Liu, X.D., Osher, S. and Chan, T., Weighted essentially non-oscillatory scheme, J. Comput. Phys., 115 (1994), 200–212.CrossRefGoogle Scholar
[19]Patera, A. T., A spectral element methd for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54 (1984), 468–488.Google Scholar
[20]Qiu, J.X. and Shu, C.W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case, J. Comput. Phys., 193(2004), 115–135.CrossRefGoogle Scholar
[21]Qiu, J.X. and Shu, C.W., Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters, SIAM J. Sci. Comput., 26 (2005), 907–929.CrossRefGoogle Scholar
[22]Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43(1981), 357–372.CrossRefGoogle Scholar
[23]Shen, Y.Q., Zha, G.C., Improvement of weighted essentially non-oscillatory schemes near discontinuities, 19th AIAA Computational Fluid Dynamics, 22 - 25, June 2009, San Antonio, Texas, AIAA 2009-3655 (2009).Google Scholar
[24]Shen, Y.Q., Zha, G.C., Improvement of weighted essentially non-oscillatory schemes near discontinuities, Computers & Fluids, 96 (2014), 1–9.CrossRefGoogle Scholar
[25]Sod, G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27 (1978), 1–31.Google Scholar
[26]Shu, C.W., High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems, SIAM Review, 51 (2009), 82–126.Google Scholar
[27]Shu, C.W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439–471.Google Scholar
[28]Shu, C.W. and Osher, O., Efficient implementation of essentially non-oscillatory shock capturing schemes, II, J. Comput. Phys., 83 (1989), 32–78.Google Scholar
[29]Spiteri, R. and Ruuth, S.J., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40 (2002), 469–491.Google Scholar
[30]Sun, Y., Wang, Z.J. and Liu, Y., High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids, Comm. in Comput. Phys., 2 (2007), 310–333.Google Scholar
[31]Titarev, V.A. and Toro, E.F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17 (2002), 609–618.Google Scholar
[32]Toro, E.F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Third edition, Springer-Verlag, Berlin, 2009.Google Scholar
[33]Toro, E.F. and Titarev, V.A., Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys., 212 (2006), 150–165.Google Scholar
[34]Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 178 (2002), 210–251.CrossRefGoogle Scholar
[35]Wang, Z.J., Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation, J. Comput. Phys., 179 (2002), 665–697.Google Scholar
[36]Wang, Z.J. and Gao, H.Y., A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys., 228 (2009), 8161–8186.Google Scholar
[37]Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (2009), 115–173.Google Scholar
[38]Xiao, F., Ii, S., Chen, C. G. and Li, X.L., A note on the general multi-moment constrained flux reconstruction formulation for high order schemes, Appl. Math. Modelling, 37 (2013), 5092–5108.CrossRefGoogle Scholar
[39]Xiao, F., Yabe, T. and Ito, T., Constructing Oscillation Preventing Scheme for Advection Equation by Rational Function, Computer Physics Communications, 93 (1996), 1–12.Google Scholar
[40]Xiao, F., Yabe, T., Nizam, G. and Ito, T., Constructing a multi-dimensional oscillation preventing scheme for the advection equation by a rational function, Computer Physics Communications, 94 (1996), 103–118.Google Scholar
[41]Xiao, F. and Yabe, T., Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation, J. Comput. Phys., 170(2001), 498–522.Google Scholar
[42]Xiao, F., Yabe, T., Peng, X.D. and Kobayashi, H., Conservative and oscillation-less atmospheric transport schemes based on rational functions, J. Geophys. Res. 107(2002), D22, ACL2-1-ACL2-11. Doi: 10.1029/2001JD001532.Google Scholar
[43]Xie, B., Ii, S., Ikebata, A. and Xiao, F., Amulti-moment finite volume method for incompressible Navier-Stokes equations on unstructured grids: volume-average/point-value formulation, J. Comput. Phys., 277 (2014), 138–162.Google Scholar
[44]Xie, B. and Xiao, F., Two and three dimensional multi-moment finite volume solver for incompressible Navier-Stokes equations on unstructured grids with arbitrary quadrilateral and hexahedral elements, Computers & Fluids, 104(2014), 40–54.Google Scholar
[45]Yabe, T. and Aoki, T., A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver, Computer Physics Communications, 66 (1991), 219–232.Google Scholar
[46]Yabe, T., Xiao, F. and Utsumi, T., The constrained interpolation profile method for multiphase analysis, J. Comput. Phys., 169(2001), 556–593.Google Scholar
[47]Yee, H., Sandham, N., and Djomehri, M., Low-dissipative high-order shock-capturing methods using characteristic-based filters, J. Comput. Phys., 150 (1999), 199–238.Google Scholar
[48]Zhong, X. and Shu, C.W., A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods, J. Comput. Phys., 232 (2013), 397–415.Google Scholar