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Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. (English) Zbl 1365.76149
Summary: This paper extends the MOOD method proposed by the authors in [J. Comput. Phys. 230, No. 10, 4028–4050 (2011; Zbl 1218.65091)], along two complementary axes: extension to very high-order polynomial reconstruction on non-conformal unstructured meshes and new detection criteria. The former is a natural extension of the previous cited work which confirms the good behavior of the MOOD method. The latter is a necessary brick to overcome limitations of the discrete maximum principle used in the previous work. Numerical results on advection problems and hydrodynamics Euler equations are presented to show that the MOOD method is effectively high-order (up to sixth-order), intrinsically positivity-preserving on hydrodynamics test cases and computationally efficient.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
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##### References:
 [1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J comput phys, 114, 45-58, (1994) · Zbl 0822.65062 [2] Abgrall, R., Essentially non-oscillatory residual distribution schemes for hyperbolic problems, J comput phys, 214, 773-808, (2006) · Zbl 1089.65083 [3] Barth TJ. Numerical methods for conservation laws on structured and unstructured meshes. VKI March 2003 lectures series. [4] Barth TJ, Fredrickson PO. Higher-order solution of the euler equations on unstructured grids using quadratic reconstruction. In: AIAA conference paper 90-0013; 1990. [5] Buffard, T.; Clain, S., Monoslope and multislope MUSCL methods for unstructured meshes, J comput phys, 229, 3745-3776, (2010) · Zbl 1189.65204 [6] Clain, S.; Clauzon, V., L∞ stability of the MUSCL methods, Numer math, 116, 31-64, (2010) · Zbl 1228.65180 [7] Clain, S.; Diot, S.; Loubère, R., A high-order finite volume method for hyperbolic systems: multi-dimensional optimal order detection (MOOD), J comput phys, 230, 10, 4028-4050, (2011) · Zbl 1218.65091 [8] Cockburn, B.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework, Math comput, 52, 411-435, (1989) · Zbl 0662.65083 [9] Cockburn, B.; Lin, S.Y.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J comput phys, 84, 90-113, (1989) · Zbl 0677.65093 [10] Cockburn, B.; Hou, S.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math comput, 54, 545-581, (1990) · Zbl 0695.65066 [11] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J comput phys, 141, 199-224, (1998) · Zbl 0920.65059 [12] Csı´k, A.; Ricchiuto, M.; Deconinck, H., A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws, J comput phys, 179, 286-312, (2002) · Zbl 1005.65111 [13] Dumbser, M.; Castro, M.; Parés, C.; Toro, E.F., ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows, Comput fluids, 38, 1731-1748, (2009) · Zbl 1177.76222 [14] Dumbser, M.; Kser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J comput phys, 221, 693-723, (2007) · Zbl 1110.65077 [15] Dumbser, M.; Kser, M.; Titarev, V.A.; Toro, E.F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J comput phys, 226, 204-243, (2007) · Zbl 1124.65074 [16] Harris, R.; Wang, Z.J.; Liu, Y., Efficient quadrature-free high-order spectral volume method on unstructured grids: theory and 2D implementation, J comput phys, 227, 1620-1642, (2008) · Zbl 1134.65070 [17] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly highorder accurate nonoscillatory schemes III, J comput phys, 71, 279-309, (1987) [18] Hu, C.; Shu, C.W., Weighted essentially non-oscillatory schemes on triangular meshes, J comput phys, 150, 97-127, (1999) · Zbl 0926.65090 [19] Hubbard, M.E., Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids, J comput phys, 155, 1, 54-74, (1999) · Zbl 0934.65109 [20] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J comput phys, 126, 202-228, (1996) · Zbl 0877.65065 [21] Jiang, G.-S.; Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservative laws, SIAM J sci comput, 19, 1892-1917, (1998) · Zbl 0914.65095 [22] Kolgan, V.P., Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics, Trans central aerohydrodynam inst, 3, 68-77, (1972), [in Russian] [23] Kolgan, V.P., Finite-difference schemes for computation of three dimensional solutions of gas dynamics and calculation of a flow over a body under an angle of attack, Trans central aerohydrodynam inst, 6, 1-6, (1975), [in Russian] [24] Kolgan, V.P., Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics, J comput phys, 230, 2384-2390, (2010) · Zbl 1316.76063 [25] Leveque, Randall J., High-resolution conservative algorithms for advection in incompressible flow, SIAM J numer anal, 33, 627-665, (1996) · Zbl 0852.76057 [26] Loubère, R.; Maire, P.-H.; Vachal, P., Staggered Lagrangian discretization based on cell-centered Riemann solver and associated hydrodynamics scheme, Commun comput phys, 10, 4, 940-978, (2011) · Zbl 1373.76138 [27] Maire, Pierre-Henri, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J comput phys, 228, 2391-2425, (2009) · Zbl 1156.76434 [28] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of Mean values on unstructured grids, J comput phys, 144, 194-212, (1998) · Zbl 1392.76048 [29] Ollivier-Gooch, C.F., Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction, J comput phys, 133, 6-17, (1997) · Zbl 0899.76282 [30] Osher, S.; Chakravarthy, S., High resolution schemes and the entropy condition, SIAM J numer anal, 21, 955-984, (1984) · Zbl 0556.65074 [31] Park, J.S.; Yoon, S.-H.; Kim, C., Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids, J comput phys, 229, 788-812, (2010) · Zbl 1185.65150 [32] Ricchiuto, M.; Bollermann, A., Stabilized residual distribution for shallow water simulations, J comput phys, 228, 1071-1115, (2009) · Zbl 1330.76097 [33] Sander, R., A third-order accurate variation non-expansive difference scheme for single nonlinear conservation law, Math comput, 51, 535-558, (1988) [34] Shi, J.; Hu, C.; Shu, C.W., A technique of treating negative weights in WENO schemes, J comput phys, 175, 108-127, (2002) · Zbl 0992.65094 [35] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111 [36] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing scheme, J comput phys, 77, 439-471, (1988) · Zbl 0653.65072 [37] Skews, B.W.; Kleine, H., Flow features resulting from shock wave impact on a cylindrical cavity, J fluid mech, 580, 481-493, (2007) · Zbl 1113.76010 [38] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional non-linear hyperbolic systems, J comput phys, 204, 715-736, (2005) · Zbl 1060.65641 [39] Toro, E.F.; Hidalgo, A., ADER finite volume schemes for nonlinear reaction – diffusion equations, Appl numer math, 59, 73-100, (2009) · Zbl 1155.65065 [40] Tsoutsanis, P.; Titarev, V.A.; Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J comput phys, 230, 1585-1601, (2011) · Zbl 1210.65160 [41] Van Leer B. Towards the ultimate conservative difference scheme I. The quest of monotonicity. In: Proceedings of the third international conference on numerical methods in fluid mechanics, Lecture notes in physics, vol. 18; 1973. p. 163-8. · Zbl 0255.76064 [42] Van Leer, B., Towards the ultimate conservative difference scheme II. monotonicity and conservation combined in a second-order scheme, J comput phys, 14, 361-370, (1974) · Zbl 0276.65055 [43] Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J comput phys, 178, 210-251, (2002) · Zbl 0997.65115 [44] Wang, Z.J.; Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids: extention to two dimensional scalar equation, J comput phys, 179, 665-697, (2002) · Zbl 1006.65113 [45] Wolf, W.R.; Azevedo, J.L.F., High-order ENO and WENO schemes for unstructured grids, Int J numer methods fluids, 55, 917-943, (2007) · Zbl 1388.76217 [46] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J comput phys, 54, 115-173, (1984) · Zbl 0573.76057 [47] Yee, H.C.; Vinokur, M.; Djomehri, M.J., Entropy splitting and numerical dissipation, J comput phys, 162, 33-81, (2000) · Zbl 0987.65086 [48] Zhang, Y.-T.; Shu, C.-W., Third-order WENO scheme on three dimensional tetrahedral meshes, Commun comput phys, 5, 836-848, (2009) · Zbl 1364.65177 [49] Zhang, X.; Shu, C.-W., Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments, Proc roy soc A, 467, 2752-2776, (2011) · Zbl 1222.65107
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