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FORCE schemes on unstructured meshes. II: Non-conservative hyperbolic systems. (English) Zbl 1227.76043
Summary: In this paper we propose a new high order accurate centered path-conservative method on unstructured triangular and tetrahedral meshes for the solution of multi-dimensional non-conservative hyperbolic systems, as they typically arise in the context of compressible multi-phase flows. Our path-conservative centered scheme is an extension of the centered method recently proposed in [E. F. Toro, A. Hidalgo and M. Dumbser, J. Comput. Phys. 228, No. 9, 3368–3389 (2009; Zbl 1168.65377)] for conservation laws, to which it reduces if the system matrix is the Jacobian of a flux function. The main advantage in the proposed centered approach compared to upwind methods is that no information about the eigenstructure of the system or Roe averages are needed. The final fully discrete high order accurate formulation in space and time is obtained using the general framework of $$P_{N}P_{M}$$ schemes proposed in [M. Dumbser et al., J. Comput. Phys. 227, No. 18, 8209–8253 (2008; Zbl 1147.65075)], which unifies in one single general family of schemes classical finite volume and discontinuous Galerkin methods. These $$P_{N}P_{M}$$ methods can also be called reconstructed discontinuous Galerkin schemes, due to the use of the $$P_{N}P_{M}$$ least-squares reconstruction operator. We show applications of our high order accurate unstructured centered method to the two- and three-dimensional Baer-Nunziato equations of compressible multiphase flows as introduced in [M. R. Baer and J. W. Nunziato [Int. J. Multiphase Flow 12, 861–889 (1986; Zbl 0609.76114)].

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 76M10 Finite element methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76T30 Three or more component flows
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##### References:
 [1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. comput. phys., 144, 45-58, (1994) · Zbl 0822.65062 [2] Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach, J. comput. phys., 125, 150-160, (1996) · Zbl 0847.76060 [3] Andrianov, N.; Warnecke, G., The Riemann problem for the baer – nunziato two-phase flow model, J. comput. phys., 212, 434-464, (2004) · Zbl 1115.76414 [4] Baer, M.R.; Nunziato, J.W., A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, J. multiphase flow, 12, 861-889, (1986) · Zbl 0609.76114 [5] Balsara, D., Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophy. J. suppl. ser., 151, 149-184, (2004) [6] T.J. Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper No. 90-0013, 28th Aerospace Sciences Meeting, January 1990. [7] Canestrelli, A.; Siviglia, A.; Dumbser, M.; Toro, E.F., A well-balanced high order centered scheme for nonconservative systems: application to shallow water flows with fix and mobile bed, Adv. water resour., 32, 834-844, (2009) [8] Castro, M.J.; Fernández, E.; Ferreiro, A.; Parés, C., Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach over unstructured meshes, Comput. meth. appl. mech. engng., 198, 2520-2538, (2009) · Zbl 1228.76091 [9] Castro, M.J.; Fernández, E.; Ferriero, A.; Garcı´a, J.A.; Parés, C., High order extensions of roe schemes for two dimensional nonconservative hyperbolic systems, J. sci. comput., 39, 67-114, (2009) · Zbl 1203.65131 [10] Castro, M.J.; Gallardo, J.M.; López, J.A.; Parés, C., Well-balanced high order extensions of godunov’s method for semilinear balance laws, SIAM J. numer. anal., 46, 1012-1039, (2008) · Zbl 1159.74045 [11] Castro, M.J.; Gallardo, J.M.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math. comput., 75, 1103-1134, (2006) · Zbl 1096.65082 [12] Castro, M.J.; LeFloch, P.G.; Muñoz-Ruiz, M.L.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. comput. phys., 227, 8107-8129, (2008) · Zbl 1176.76084 [13] M.J. Castro, A. Pardo, C. Parés, E.F. Toro, On some fast well-balanced first order solvers for nonconservative systems, Mathematics of Computation, in press. · Zbl 1369.65107 [14] 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 [15] Deledicque, V.; Papalexandris, M.V., An exact Riemann solver for compressible two-phase flow models containing non-conservative products, J. comput. phys., 222, 217-245, (2007) · Zbl 1216.76044 [16] Dumbser, M.; Balsara, D.; Toro, E.F.; Munz, C.D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes, J. comput. phys., 227, 8209-8253, (2008) · Zbl 1147.65075 [17] Dumbser, M.; Castro, M.; Parés, C.; Toro, E.F., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Comp. fluid., 38, 1731-1748, (2009) · Zbl 1177.76222 [18] Dumbser, M.; Enaux, C.; Toro, E.F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. comput. phys., 227, 3971-4001, (2008) · Zbl 1142.65070 [19] Dumbser, M.; Käser, 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 [20] Dumbser, M.; Käser, 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 [21] Dumbser, M.; Munz, C.D., Building blocks for arbitrary high order discontinuous Galerkin schemes, J. sci. comput., 27, 215-230, (2006) · Zbl 1115.65100 [22] Gallardo, J.M.; Parés, C.; Castro, M.J., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. comput. phys., 227, 574-601, (2007) · Zbl 1126.76036 [23] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067 [24] Hu, C.; Shu, C.W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97-127, (1999) · Zbl 0926.65090 [25] 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. cent. aerohydrodynam. inst., 3, 6, 68-77, (1972), (in Russian) [26] Dal Maso, G.; LeFloch, P.G.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068 [27] Muñoz, M.L.; Parés, C., Godunov method for nonconservative hyperbolic systems, Math. model. numer. anal., 41, 169-185, (2007) · Zbl 1124.65077 [28] Murrone, A.; Guillard, H., A five equation reduced model for compressible two phase flow problems, J. comput. phys., 202, 664-698, (2005) · Zbl 1061.76083 [29] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. numer. anal., 44, 300-321, (2006) · Zbl 1130.65089 [30] Rhebergen, S.; Bokhove, O.; van der Vegt, J.J.W., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. comput. phys., 227, 1887-1922, (2008) · Zbl 1153.65097 [31] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066 [32] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. comput. phys., 150, 425-467, (1999) · Zbl 0937.76053 [33] Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. sci. comput., 21, 1115-1145, (1999) · Zbl 0957.76057 [34] Schwendeman, D.W.; Wahle, C.W.; Kapila, A.K., The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J. comput. phys., 212, 490-526, (2006) · Zbl 1161.76531 [35] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, NJ · Zbl 0379.65013 [36] Suresh, A.; Huynh, H.T., Accurate monotonicity-preserving schemes with runge – kutta time stepping, J. comput. phys., 136, 83-99, (1997) · Zbl 0886.65099 [37] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. comput. phys., 204, 715-736, (2005) · Zbl 1060.65641 [38] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer · Zbl 0923.76004 [39] Toro, E.F.; Billet, S.J., Centered TVD schemes for hyperbolic conservation laws, IMA J. numer. anal., 20, 44-79, (2000) · Zbl 0943.65100 [40] Toro, E.F.; Hidalgo, A.; Dumbser, M., FORCE schemes on unstructured meshes I: conservative hyperbolic systems, J. comput. phys., 228, 3368-3389, (2009) · Zbl 1168.65377 [41] Toumi, I., A weak formulation of roe’s approximate Riemann solver, J. comput. phys., 102, 360-373, (1992) · Zbl 0783.65068 [42] van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223 [43] 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 [44] 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
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