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

Berthon, Christophe; Coudière, Yves; Desveaux, Vivien

ESAIM: Mathematical Modelling and Numerical Analysis , Multiscale problems and techniques. Special Issue, Tome 48 (2014) no. 2, pp. 583-602.

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.

[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. | MR | Zbl

[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. | MR | Zbl

[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] 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] C. Berthon, Stability of the MUSCL schemes for the Euler equations. Commun. Math. Sci. 3 (2005) 133-158. | MR | Zbl

[6] C. Berthon, Numerical approximations of the 10-moment Gaussian closure. Math. Comput. 75 (2006) 1809-1832. | MR | Zbl

[7] C. Berthon, Robustness of MUSCL schemes for 2D unstructured meshes. J. Comput. Phys. 218 (2006) 495-509. | MR | Zbl

[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). | MR | Zbl

[9] F. Bouchut, C. Bourdarias and B. Perthame, A MUSCL method satisfying all the numerical entropy inequalities. Math. Comput. 65 (1996) 1439-1462. | MR | Zbl

[10] T. Buffard and S. Clain, Monoslope and multislope MUSCL methods for unstructured meshes. J. Comput. Phys. 229 (2010) 3745-3776. | MR | Zbl

[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. | MR

[12] C. Calgaro, E. Creusé, T. Goudon and Y. Penel, Positivity-preserving schemes for Euler equations: sharp and practical CFL conditions (2012). preprint. | MR | Zbl

[13] S. Clain and V. Clauzon, L∞ stability of the MUSCL methods. Numerische Mathematik 116 (2010) 31-64. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[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. | Zbl

[19] M.S. Darwish and F. Moukalled, Tvd schemes for unstructured grids. International Journal of Heat and Mass Transfer 46 (2003) 599-611. | Zbl

[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. Fluids 64 (2012) 43-63. | MR

[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. | Numdam | MR | Zbl

[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 Fluids 59 (2009) 423-442. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[25] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481-499. | MR | Zbl

[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. | MR | Zbl

[27] M.E. Hubbard, Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155 (1999) 54-74. | MR | Zbl

[28] B. Keen and S. Karni, A second order kinetic scheme for gas dynamics on arbitrary grids. J. Comput. Phys. 205 (2005) 108-130. | MR | Zbl

[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] 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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[33] R.J. Leveque, Finite volume methods for hyperbolic problems. Cambridge Univ Press (2002). | MR | Zbl

[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. | MR | Zbl

[35] Q. Liang and F. Marche, Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in Water Resources 32 (2009) 873-884.

[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. | MR | Zbl

[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).

[38] B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. (1992) 1-19. | MR | Zbl

[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. | MR | Zbl

[40] B. Perthame and C.W. Shu, On positivity preserving finite volume schemes for Euler equations. Numerische Mathematik 73 (1996) 119-130. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[43] E.F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Verlag (2009). | MR | Zbl

[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. | MR | Zbl

[45] V. Venkatakrishnan, Convergence to steady state solutions of the euler equations on unstructured grids with limiters. J. Comput. Phys. 118 (1995) 120-130. | Zbl

[46] P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54 (1984) 115-173. | MR | Zbl