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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)].

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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