Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T04:58:44.649Z Has data issue: false hasContentIssue false
  • English
  • Français

A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art

Published online by Cambridge University Press:  20 August 2015

Remi Abgrall
Remi Abgrall*
Affiliation:
Team Bacchus, Institut de Mathématiques de Bordeaux, INRIA and University of Bordeaux, 33 405 Talence cedex, France
*
*Corresponding author.Email:[email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives.

Keywords

65M6076M3076M12Hyperbolic problemsNavier Stokes equationsunstructured meshesresidual distribution methods
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 4 , April 2012 , pp. 1043 - 1080
Copyright
Copyright © Global Science Press Limited 2012

References

[1]Abgrall, R.Toward the ultimate conservative scheme: Following the quest. J. Comput. Phys., 167(2):277–315, 2001.CrossRefGoogle Scholar
[2]Abgrall, R.A residual distribution method using discontinuous elements for the computation of possibly non smooth flows. Adv. Appl. Math. Mech., 2(1):32–44, 2010.Google Scholar
[3]Abgrall, R., Andrianov, N., and Mezine, M.Towards very high-order accurate schemes for unsteady convection problems on unstructured meshes. Int. J. Numer. Methods Fluids, 47(8-9):679–691, 2005.CrossRefGoogle Scholar
[4]Abgrall, R., Huart, R., and Ricchiuto, M.Approximation of the ideal mhd equations using residual distribution methods. in preparation, 2010.Google Scholar
[5]Abgrall, R., Larat, A., Ricchiuto, M., and Tavé, C.A simple construction of very high order non-oscillatory compact schemes on unstructured meshes. Computers and Fluids, 38(7):1314–1323, 2009.Google Scholar
[6]Abgrall, R. and Roe, P. L.High-order fluctuation schemes on triangular meshes. J. Sci. Comput., 19(1-3):3–36, 2003.Google Scholar
[7]Abgrall, R. and Shu, C.-W.Development of residual distribution schemes for the discontinuous galerkin method: The scalar case with linear elements. Commun. Comput. Phys., 5:376–390, 2009.Google Scholar
[8]Abgrall, R. and Treflick, J.An example of high order residual distribution scheme using non-Lagrange elements. J. Sci. Comput., 45:3–25, 2010.CrossRefGoogle Scholar
[9]Abgrall, R.Essentially non oscillatory residual distribution schemes for hyperbolic problems. J. Comput. Phys., 214(2):773–808, 2006.CrossRefGoogle Scholar
[10]Abgrall, R.Residual distribution schemes: current status and future trends. Comput. Fluids, 35(7):641–669, 2006.Google Scholar
[11]Abgrall, R. and Mezine, M.Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems. J. Comput. Phys., 188(1):16–55, 2003.CrossRefGoogle Scholar
[12]Caraeni, D. PhD thesis, Lund University, 2002.Google Scholar
[13]Caraeni, D. and Fuchs, L.Compact third-order multidimensional upwind scheme for Navier Stokes simulations. Theor. Comput. Fluid Dynam., 15:373–401, 2002.Google Scholar
[14]Cockburn, B. and Shu, C.-W.The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 35(6):2440–2463, 1998.CrossRefGoogle Scholar
[15]Deconinck, H., Roe, P.L., and Struijs, R.A multidimensional generalisation of Roe’s difference splitter for the Euler equations. Comput. Fluids, 22(2/3):215–222, 1993.Google Scholar
[16]Deconinck, H., Struijs, R., Bourgeois, G., and Roe, P.L.Compact advection schemes on unstructured meshes. VKI Lecture Series 1993-04, Computational Fluid Dynamics, 1993.Google Scholar
[17]Hubabrd, M. and Ricchiuto, M.Discontinuous upwind residual distribution: A route to unconditional positivity and high order accuracy. Comput. Fluids, 46(1):263–269, 2011.Google Scholar
[18]Hubbard, M.Discontinuous fluctuation distribution. J. Comput. Phys., 227(24):10125–10147, 2008.Google Scholar
[19]Hughes, Th.J.R., Franca, L.P., and Mallet, M.Finite element formulation for computational fluid dynamics: I symmetric forms of the compressible Euler and Navier Stokes equations and the secoond law of thermodynamics. Comput. Meth. Appl. Mech. Engr., 54:223–234, 1986.CrossRefGoogle Scholar
[20]Hughes, Th.J.R., Mallet, M., and Mizukami, A.A new finite element formulation for computational fluid dynamics: II Beyong SUPG. Comput. Meth. Appl. Mech. Engr., 54:341–355, 1986.Google Scholar
[21]Hughes, Th.J.R. and Mallet, M.A new finite element formulation for computational fluid dynamics. IV: A discontinuity-capturing operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Engr., 58:329–336, 1986.CrossRefGoogle Scholar
[22]Johnson, C., Nävert, U., and Pitkäranta, J.Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engr., 45:285–312, 1984.CrossRefGoogle Scholar
[23]Larat, A.Conception et Analyse de Schémas Distribuant le Résidu d’Ordre Trés Elevé. Application à la Mécanique des Fluides. PhD thesis, Université de Bordeaux, 2009. http://tel.archives-ouvertes.fr/tel-00502429/fr/.Google Scholar
[24]Maerz, J. and Degrez, G.Improving the time accuracy of residual distribution schemes. Technical Report VKI-PR 96-17, von Karman Institute, 1996.Google Scholar
[25]Ni, R.-H.A multiple grid scheme for solving the Euler equations. AIAA J., 20:1565–1571, 1981.Google Scholar
[26]Nishikawa, H.A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes. J. Comput. Phys., 227(1):315–352, 2007.Google Scholar
[27]Nishikawa, H.A first-order system approach for diffusion equation. II: Unification of advection and diffusion. J. Comput. Phys., 229(11):3989–4016, 2010.CrossRefGoogle Scholar
[28]Paillère, H.Multidimensional Upwind residual Discretisation Schemes for the Euler and Navier Stokes Equations on Unstructured Meshes. PhD thesis, Université Libre de Bruxelles, 1995.Google Scholar
[29]De Palma, P., Pascazio, G., Rossiello, G., and Napolitano, M.A second-order-accurate monotone implicit fluctuation splitting scheme for unsteady problems. J. Comput. Phys., 208(1):1–33, 2005.Google Scholar
[30]Larat, A., Abgrall, R. and Ricchiuto, M.Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes. J. Comput. Phys., 230(11):4103–4136, 2011.Google Scholar
[31]Ricchiuto, M. and Abgrall, R.Explicit runge-kutta residual distribution schemes for time dependent problems: Second order case. J. Comput. Phys., 229(16):5653–5691, 1ugust 2010.Google Scholar
[32]Ricchiuto, M., Villedieu, N., Abgrall, R., and Deconinck, H.On uniformly high-order accurate residual distribution schemes for advection-diffusion. J. Comput. Appl. Math., 215(2):547–556, 2008.Google Scholar
[33]Ricchiuto, M. and Bollermann, A.Stabilized residual distribution for shallow water simulations. J. Comput. Phys., 228(4):1071–1115, 2009.Google Scholar
[34]Ricchiuto, M., Csík, A., and Deconinck, H.Residual distribution for general time-dependent conservation laws. J. Comput. Phys., 209(1):249–289, 2005.CrossRefGoogle Scholar
[35]Roe, P. L.Characteristic-based schemes for the Euler equations. Annu. Rev. Fluid Mech., 18:337–365, 1986.Google Scholar
[36]Roe, P.L.Approximate riemann solver, parameter vectors and difference schemes. J. Comput. Phys., 43:357–372, 1981.Google Scholar
[37]Roe, P.L. and Sidilkover, D.Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal., 29(6):1542–1568, 1992.CrossRefGoogle Scholar
[38]Rossiello, G., De Palma, P., Pascazio, G., and Napolitano, M.Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems. J. Comput. Phys., 222(1):332–352, 2007.Google Scholar
[39]Struijs, R., Deconinck, H., and Roe, P. L.Fluctuation Splitting Schemes for the 2D Euler equations. VKI LS 1991-01, Computational Fluid Dynamics, 1991.Google Scholar
[40]Struijs, R., Deconinck, H., and Roe, P.L.Fluctuation splitting schemes for the 2D Euler equations. VKI LS 1991-01, Computational Fluid Dynamics, 1991.Google Scholar
[41]van Leer, B.Towards the ultimate conservative difference scheme. IV: A new approach to numerical convection. J. Comput. Phys., 23:276–299, 1977.CrossRefGoogle Scholar
[42]Villedieu, N., Quintino, T., Ricchiuto, M., and Deconinck, H.Third order residual distribution scheme for the Navier Stokes equations. J. Comput. Phys., 230(11):4301–4315, 2010.Google Scholar
[43]Woodward, P. and Colella, P.The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54:115–173, 1984.Google Scholar