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

Christophe Berthon; Yves Coudière; Vivien Desveaux

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 2, page 583-602
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Berthon, Christophe, Coudière, Yves, and Desveaux, Vivien. "Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 583-602. <http://eudml.org/doc/273187>.

@article{Berthon2014,
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.},
author = {Berthon, Christophe, Coudière, Yves, Desveaux, Vivien},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {systems of conservation laws; muscl method; unstructured meshes; dual mesh; invariant region; monotone upwind schemes for conservation laws (MUSCL); Courant-Friedrichs-Lewy (CFL) conditions; gradient reconstruction; numerical experiment},
language = {eng},
number = {2},
pages = {583-602},
publisher = {EDP-Sciences},
title = {Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)},
url = {http://eudml.org/doc/273187},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Berthon, Christophe
AU - Coudière, Yves
AU - Desveaux, Vivien
TI - Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 2
SP - 583
EP - 602
AB - 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.
LA - eng
KW - systems of conservation laws; muscl method; unstructured meshes; dual mesh; invariant region; monotone upwind schemes for conservation laws (MUSCL); Courant-Friedrichs-Lewy (CFL) conditions; gradient reconstruction; numerical experiment
UR - http://eudml.org/doc/273187
ER -

References

top
  1. [1] B. Andreianov, M. Bendahmane and K.H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations. J. Hyperbolic Differ. Equ.7 (2010) 1–67. Zbl1207.35020MR2646796
  2. [2] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ.23 (2007) 145–195. Zbl1111.65101MR2275464
  3. [3] 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). 
  4. [4] 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). 
  5. [5] C. Berthon, Stability of the MUSCL schemes for the Euler equations. Commun. Math. Sci.3 (2005) 133–158. Zbl1161.65344MR2164194
  6. [6] C. Berthon, Numerical approximations of the 10-moment Gaussian closure. Math. Comput.75 (2006) 1809–1832. Zbl1105.76036MR2240636
  7. [7] C. Berthon, Robustness of MUSCL schemes for 2D unstructured meshes. J. Comput. Phys.218 (2006) 495–509. Zbl1161.65345MR2269374
  8. [8] 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). Zbl1086.65091MR2128209
  9. [9] F. Bouchut, C. Bourdarias and B. Perthame, A MUSCL method satisfying all the numerical entropy inequalities. Math. Comput.65 (1996) 1439–1462. Zbl0853.65091MR1348038
  10. [10] T. Buffard and S. Clain, Monoslope and multislope MUSCL methods for unstructured meshes. J. Comput. Phys.229 (2010) 3745–3776. Zbl1189.65204MR2609751
  11. [11] C. Calgaro, E. Chane-Kane, E. Creusé and T. Goudon, 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. Zbl05784788MR2657857
  12. [12] C. Calgaro, E. Creusé, T. Goudon and Y. Penel, Positivity-preserving schemes for Euler equations: sharp and practical CFL conditions (2012). preprint. Zbl1284.65110MR2999785
  13. [13] S. Clain and V. Clauzon, L∞ stability of the MUSCL methods. Numerische Mathematik116 (2010) 31–64. Zbl1228.65180MR2660445
  14. [14] 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). Zbl1218.65091MR2783831
  15. [15] F. Coquel and B. Perthame, Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics. SIAM J. Numer. Anal.35 (1998) 2223–2249. Zbl0960.76051MR1655844
  16. [16] Y. Coudière and F. Hubert, A 3d discrete duality finite volume method for nonlinear elliptic equations. SIAM J. Sci. Comput.33 (2011) 1739–1764. Zbl1243.35061MR2831032
  17. [17] Y. Coudière, C. Pierre, O. Rousseau and R. Turpault, A 2D/3D discrete duality finite volume scheme. Application to ECG simulation. Int. J. Finite 6 (2009) 24. MR2500950
  18. [18] P.H. Cournède, B. Koobus and A. Dervieux, 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. Zbl1208.76088
  19. [19] M.S. Darwish and F. Moukalled, Tvd schemes for unstructured grids. International Journal of Heat and Mass Transfer46 (2003) 599–611. Zbl1121.76357
  20. [20] S. Diot, S. Clain, R. Loubère, Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids64 (2012) 43–63. Zbl1246.76073MR2982757
  21. [21] K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. Math. Model. Numer. Anal.39 (2005) 1203–1249. Zbl1086.65108MR2195910
  22. [22] R. Ghostine, G. Kesserwani, R. Mosé, J. Vazquez and A. Ghenaim, An improvement of classical slope limiters for high-order discontinuous Galerkin method. Internat. J. Numer. Methods Fluids59 (2009) 423–442. Zbl1189.65224MR2488296
  23. [23] 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). Zbl0860.65075MR1410987
  24. [24] 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. Zbl0565.65051MR693713
  25. [25] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys.160 (2000) 481–499. Zbl0949.65101MR1763823
  26. [26] F. Hermeline, 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. Zbl1173.76362MR2319051
  27. [27] M.E. Hubbard, Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys.155 (1999) 54–74. Zbl0934.65109MR1716501
  28. [28] B. Keen and S. Karni, A second order kinetic scheme for gas dynamics on arbitrary grids. J. Comput. Phys.205 (2005) 108–130. Zbl1087.76088MR2132305
  29. [29] K. Kitamura and E. Shima, Simple and parameter-free second slope limiter for unstructured grid aerodynamic simulations. AIAA J.50 (2012) 1415–1426. 
  30. [30] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Equ.18 (2002) 584–608. Zbl1058.76046MR1919599
  31. [31] 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. Zbl0268.35014MR393870
  32. [32] 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). Zbl0268.35062MR350216
  33. [33] R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Univ Press (2002). Zbl1010.65040MR1925043
  34. [34] 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. Zbl1252.65151MR2825719
  35. [35] Q. Liang and F. Marche, Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in Water Resources32 (2009) 873–884. 
  36. [36] 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. Zbl0791.65068MR1220647
  37. [37] 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). Zbl1287.76171
  38. [38] B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. (1992) 1–19. Zbl0744.76088MR1149081
  39. [39] B. Perthame and Y. Qiu, A variant of Van Leer’s method for multidimensional systems of conservation laws. J. Computat. Phys.112 (1994) 370–381. Zbl0816.65055MR1277283
  40. [40] B. Perthame and C.W. Shu, On positivity preserving finite volume schemes for Euler equations. Numerische Mathematik73 (1996) 119–130. Zbl0857.76062MR1379283
  41. [41] J. Shi, Y.T. Zhang and C.W. Shu, Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys.186 (2003) 690–696. Zbl1047.76081MR1973202
  42. [42] 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. Zbl0927.65111MR1728856
  43. [43] E.F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Verlag (2009). Zbl1227.76006MR2731357
  44. [44] B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32 (1979) 101–136. Zbl0939.76063MR1703646
  45. [45] V. Venkatakrishnan, Convergence to steady state solutions of the euler equations on unstructured grids with limiters. J. Comput. Phys.118 (1995) 120–130. Zbl0858.76058
  46. [46] P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys.54 (1984) 115–173. Zbl0573.76057MR748569

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.