Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T08:15:44.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)

Published online by Cambridge University Press:  20 January 2014

Christophe Berthon ,
Yves Coudière  and
Vivien Desveaux
Christophe Berthon
Affiliation:
Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France.. vivien.desveaux@univ-nantes.fr
Yves Coudière
Affiliation:
Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France. INRIA Sud-Ouest, 351 cours de la Libération, 33405 Talence Cedex, France.
Vivien Desveaux
Affiliation:
Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France.. vivien.desveaux@univ-nantes.fr
Get access

Abstract

We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.

Keywords

Systems of conservation lawsMUSCL methodunstructured meshesdual meshinvariant region
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 583 - 602
Copyright
© EDP Sciences, SMAI, 2014

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

Andreianov, B., Bendahmane, M. and Karlsen, K.H., Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations. J. Hyperbolic Differ. Equ. 7 (2010) 1–67. Google Scholar
Andreianov, B., Boyer, F. and Hubert, F., Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145–195. Google Scholar
T. Barth and D. Jespersen, The design and application of upwind schemes on unstructured meshes, in AIAA, Aerospace Sciences Meeting, 27 th, Reno, NV (1989).
M. Berger, M.J. Aftosmis and S.M. Murman, Analysis of slope limiters on irregular grids, in 43rd AIAA Aerospace Sciences Meeting, volume NAS Technical Report NAS-05-007 (2005).
Berthon, C., Stability of the MUSCL schemes for the Euler equations. Commun. Math. Sci. 3 (2005) 133–158. Google Scholar
Berthon, C., Numerical approximations of the 10-moment Gaussian closure. Math. Comput. 75 (2006) 1809–1832. Google Scholar
Berthon, C., Robustness of MUSCL schemes for 2D unstructured meshes. J. Comput. Phys. 218 (2006) 495–509. Google Scholar
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004).
Bouchut, F., Bourdarias, C. and Perthame, B., A MUSCL method satisfying all the numerical entropy inequalities. Math. Comput. 65 (1996) 1439–1462. Google Scholar
Buffard, T. and Clain, S., Monoslope and multislope MUSCL methods for unstructured meshes. J. Comput. Phys. 229 (2010) 3745–3776. Google Scholar
Calgaro, C., Chane-Kane, E., Creusé, E. and Goudon, T., L∞-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios. J. Comput. Phys. 229 (2010) 6027–6046. Google Scholar
C. Calgaro, E. Creusé, T. Goudon and Y. Penel, Positivity-preserving schemes for Euler equations: sharp and practical CFL conditions (2012). preprint.
Clain, S. and Clauzon, V., L∞ stability of the MUSCL methods. Numerische Mathematik 116 (2010) 31–64. Google Scholar
S. Clain, S. Diot and R. Loubère, A high-order finite volume method for hyperbolic systems: Multi-dimensional Optimal Order Detection (MOOD). J. Comput. Phys. (2011).
Coquel, F. and Perthame, B., Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics. SIAM J. Numer. Anal. 35 (1998) 2223–2249. Google Scholar
Coudière, Y. and Hubert, F., A 3d discrete duality finite volume method for nonlinear elliptic equations. SIAM J. Sci. Comput. 33 (2011) 1739–1764. Google Scholar
Coudière, Y., Pierre, C., Rousseau, O. and Turpault, R., A 2D/3D discrete duality finite volume scheme. Application to ECG simulation. Int. J. Finite 6 (2009) 24. Google Scholar
Cournède, P.H., Koobus, B. and Dervieux, A., Positivity statements for a mixed-element-volume scheme on fixed and moving grids. European J. Comput. Mechanics/Revue Européenne de Mécanique Numérique 15 (2006) 767–798. Google Scholar
Darwish, M.S. and Moukalled, F., Tvd schemes for unstructured grids. International Journal of Heat and Mass Transfer 46 (2003) 599–611. Google Scholar
Diot, S., Clain, S., Loubère, R., Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids 64 (2012) 43–63. Google Scholar
Domelevo, K. and Omnes, P., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. Math. Model. Numer. Anal. 39 (2005) 1203–1249. Google Scholar
Ghostine, R., Kesserwani, G., Mosé, R., Vazquez, J. and Ghenaim, A., An improvement of classical slope limiters for high-order discontinuous Galerkin method. Internat. J. Numer. Methods Fluids 59 (2009) 423–442. Google Scholar
E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, in vol. 118 of Appl. Math. Sci. Springer-Verlag, New York (1996).
A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review (1983) 35–61.
Hermeline, F., A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481–499. Google Scholar
Hermeline, F., Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2497–2526. Google Scholar
Hubbard, M.E., Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155 (1999) 54–74. Google Scholar
Keen, B. and Karni, S., A second order kinetic scheme for gas dynamics on arbitrary grids. J. Comput. Phys. 205 (2005) 108–130. Google Scholar
Kitamura, K. and Shima, E., Simple and parameter-free second slope limiter for unstructured grid aerodynamic simulations. AIAA J. 50 (2012) 1415–1426. Google Scholar
Kurganov, A. and Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Equ. 18 (2002) 584–608. Google Scholar
P.D. Lax, Shock waves and entropy, in Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971). Academic Press, New York (1971) 603–634.
P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conference Board of the Math. Sci. Regional Conference Series Appl. Math. No. 11. Society for Industrial and Applied Mathematics, Philadelphia, Pa. (1973).
R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Univ Press (2002).
Wanai Li, Yu-Xin Ren, Guodong Lei and Hong Luo, The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids. J. Comput. Phys. 230 (2011) 7775–7795.
Liang, Q. and Marche, F., Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in Water Resources 32 (2009) 873–884. Google Scholar
X.D. Liu, A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws. SIAM J. Numer. Anal. (1993) 701–716.
K. Michalak and C. Ollivier-Gooch, Limiters for unstructured higher-order accurate solutions of the euler equations, in Proc. of the AIAA Forty-sixth Aerospace Sciences Meeting (2008).
B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. (1992) 1–19.
Perthame, B. and Qiu, Y., A variant of Van Leer’s method for multidimensional systems of conservation laws. J. Computat. Phys. 112 (1994) 370–381. Google Scholar
Perthame, B. and Shu, C.W., On positivity preserving finite volume schemes for Euler equations. Numerische Mathematik 73 (1996) 119–130. Google Scholar
Shi, J., Zhang, Y.T. and Shu, C.W., Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys. 186 (2003) 690–696. Google Scholar
C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced numerical approximation nonlinear Hyperbolic equations (1998) 325–432.
E.F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Verlag (2009).
Van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32 (1979) 101–136. Google Scholar
Venkatakrishnan, V., Convergence to steady state solutions of the euler equations on unstructured grids with limiters. J. Comput. Phys. 118 (1995) 120–130. Google Scholar
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