A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems

Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125, 150–160 (1996)

Abgrall, R., Karni, S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759–2763 (2010)

Andrianov, N., Warnecke, G.: The Riemann problem for the Baer–Nunziato two-phase flow model. J. Comput. Phys. 212, 434–464 (2004)

Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)

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)

Bernetti, R., Titarev, V.A., Toro, E.F.: Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry. J. Comput. Phys. 227, 3212–3243 (2008)

Canestrelli, A., Dumbser, M., Siviglia, A., Toro, E.F.: Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed. Adv. Water Res. 33, 291–303 (2010)

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 Res. 32, 834–844 (2009)

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)

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)

Castro, M.J., Pardo, A., Parés, C., Toro, E.F.: On some fast well-balanced first order solvers for nonconservative systems. Math. Comput. 79, 1427–1472 (2010)

Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

Deledicque, V., Papalexandris, M.V.: An exact Riemann solver for compressible two-phase flow models containing non-conservative products. J. Comput. Physics 222, 217–245 (2007)

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)

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)

Dumbser, M., Hidalgo, A., Castro, M., Parés, C., Toro, E.F.: FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng. 199, 625–647 (2010)

Ferrari, A.: SPH simulation of free surface flow over a sharp-crested weir. Adv. Water Res. 33, 270–276 (2010)

Ferrari, A., Dumbser, M., Toro, E.F., Armanini, A.: A new stable version of the SPH method in Lagrangian coordinates. Commun. Comput. Phys. 4, 378–404 (2008)

Ferrari, A., Dumbser, M., Toro, E.F., Armanini, A.: A new 3D parallel SPH scheme for free surface flows. Comput. Fluids 38, 1203–1217 (2009)

Garcia-Navarro, P., Vázquez-Cendón, M.E.: On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29, 951–979 (2000)

Glimm, J.: Solution in the large for nonlinear hyperbolic systems of equations. Commun. Pure. Appl. Math. 18, 697–715 (1965)

Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)

Hou, T.Y., LeFloch, P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62, 497–530 (1994)

Jiang, G.-S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

LeVeque, R.J.: Balancing source terms and flux gradients in highresolution Godunov methods. J. Comput. Phys. 146, 346–365 (1998)

Maso, G. Dal, LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)

Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)

Osher, S.: Riemann solvers, the entropy condition and difference approximations. SIAM J. Numer. Anal. 21, 217–235 (1984)

Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic conservation laws. Math. Comput. 38, 339–374 (1982)

Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321 (2006)

Pelanti, M., Bouchut, F., Mangeney, A.: A Roe-Type scheme for two-phase shallow granular flows over variable topography. Math. Model. Numer. Analysis 42, 851–885 (2008)

Pitman, E.B., Le, L.: A two-fluid model for avalanche and debris flows. Phil. Trans. R. Soc. A 363, 1573–1601 (2005)

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)

Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)

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)

Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)

Titarev, V.A., Toro, E.F.: ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys. 204, 715–736 (2005)

Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd edn. Springer, Berlin (1999)

Toumi, I.: A weak formulation of Roe’s approximate Riemann solver. J. Comput. Phys. 102, 360–373 (1992)

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)