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ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows. (English) Zbl 1177.76222
Summary: We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step \(P_NP_M\) schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [M. Dumbser et al., J. Comput. Phys. 227, No. 18, 8209–8253 (2008; Zbl 1147.65075)]. The two key ingredients of our high order approach are: first, the high order accurate \(P_NP_M\) reconstruction operator on unstructured meshes, using the WENO strategy presented in [M. Dumbser et al., J. Comput. Phys. 226, No. 1, 204–243 (2007; Zbl 1124.65074)] to ensure monotonicity at discontinuities, and second, a local space-time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [M. Dumbser et al., J. Comput. Phys. 227, No. 8, 3971–4001 (2008; Zbl 1142.65070)]. These two key ingredients are combined with the recently developed path-conservative methods of C. Parés [SIAM J. Numer. Anal. 44, No. 1, 300–321 (2006; Zbl 1130.65089)] and M. J. Castro et al. [Math. Comput. 75, No. 255, 1103–1134 (2006; Zbl 1096.65082)] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of E. B. Pitman and L. Le [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 363, No. 1832, 1573–1601 (2005; Zbl 1152.86302)].

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76B70 Stratification effects in inviscid fluids
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