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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)].

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
Full Text: DOI
[1] Balsara, D., Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys J suppl ser, 151, 149-184, (2004)
[2] Barth TJ, Frederickson PO. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA paper no. 90-0013, 28th Aerospace Sciences Meeting; January 1990.
[3] Bermúdez, A.; Vázquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput fluids, 23, 1049-1071, (1994) · Zbl 0816.76052
[4] Canestrelli A, Siviglia A, Dumbser M, Toro EF. A well-balanced high order centered scheme for nonconservative systems: application to shallow water flows with fix and mobile bed. Adv Water Resour 2009, in press. doi:10.1016/j.advwatres.2009.02.006.
[5] Castro MJ, 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 Eng 2009, in press.
[6] Castro, M.J.; Fernández, E.; Ferriero, A.; García, 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
[7] 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
[8] 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
[9] 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
[10] Castro MJ, Pardo A, Parés C, Toro EF. On some fast well-balanced first order solvers for nonconservative systems. Math Comput, accepted for publication.
[11] de la Puente, J.; Dumbser, M.; Käser, M.; Igel, H., Discontinuous Galerkin methods for wave propagation in poroelastic media, Geophysics, 73, T77-T97, (2008)
[12] 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
[13] 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
[14] 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
[15] 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
[16] Dumbser, M.; Moschetta, J.M.; Gressier, J., A matrix stability analysis of the carbuncle phenomenon, J comput phys, 197, 647-670, (2004) · Zbl 1079.76607
[17] Dumbser, M.; Munz, C.D., Building blocks for arbitrary high order discontinuous Galerkin schemes, J sci comput, 27, 215-230, (2006) · Zbl 1115.65100
[18] 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
[19] 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
[20] Hu, C.; Shu, C.W., Weighted essentially non-oscillatory schemes on triangular meshes, J comput phys, 150, 97-127, (1999) · Zbl 0926.65090
[21] LeFloch PG. Shock waves for nonlinear hyperbolic systems in nonconservative form. Institute for Math. and its Appl., Minneapolis, Preprint 593.
[22] LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wavepropagation algorithm, J comput phys, 146, 346-365, (1998) · Zbl 0931.76059
[23] 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
[24] Muñoz, M.L.; Parés, C., Godunov method for nonconservative hyperbolic systems, Math model numer anal, 41, 169-185, (2007) · Zbl 1124.65077
[25] Noether E. Invariante Variations probleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, 1918, pp. 235-257. · JFM 46.0770.01
[26] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J numer anal, 44, 300-321, (2006) · Zbl 1130.65089
[27] Parés, C.; Castro, M.J., On the well-balance property of roe’s method for nonconservative hyperbolic systems. applications to shallow-water systems, Math model numer anal, 38, 821-852, (2004) · Zbl 1130.76325
[28] Pelanti, M.; Bouchut, F.; Mangeney, A., A roe-type scheme for two-phase shallow granular flows over variable topography, Math model numer anal, 42, 851-885, (2008) · Zbl 1391.76801
[29] Pitman, E.B.; Le, L., A two-fluid model for avalanche and debris flows, Philos trans roy soc A, 363, 1573-1601, (2005) · Zbl 1152.86302
[30] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in Fortran 77, vol. 1, (1996), Cambridge University Press Cambridge (MA) · Zbl 0878.68049
[31] Quirk, J., A contribution to the great Riemann solver debate, Int J numer meth fluids, 18, 555-574, (1994) · Zbl 0794.76061
[32] Rhebergen, S.; Bokhove, O.; vander 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
[33] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Englewood Cliffs (NJ) · Zbl 0379.65013
[34] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J comput phys, 204, 715-736, (2005) · Zbl 1060.65641
[35] 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
[36] Toumi, I., A weak formulation of roe’s approximate Riemann solver, J comput phys, 102, 360-373, (1992) · Zbl 0783.65068
[37] van der Vegt, J.J.W.; vander Ven, H., Space – time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. general formulation, J comput phys, 182, 546-585, (2002) · Zbl 1057.76553
[38] van der Ven, H.; van der Vegt, J.J.W., Space – time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. efficient flux quadrature, Comput meth appl mech eng, 191, 4747-4780, (2002) · Zbl 1099.76521
[39] 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
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