A Kinetic Flux-vector Splitting Scheme for Two-layer Shallow Flow Model

[1] Castro, M. J., Garc´ıa-Rodr´ıguez, J. A., Gonz´alez-Vida, J. M., Mac´ıas, J., & Par´es, C., Improved FVM for two-layer shallow-water models: Application to the Strait of Gibraltar, Advances in Engineering Software, 38(6), 2007, 386-398.

[2] Ostapenko, V. V., Numerical simulation of wave flows caused by a shoreside landslide, Journal of Applied Mechanics and Technical Physics, 40(4), 1999, 647-654.

[3] Fern´andez-Nieto, E. D., Bouchut, F., Bresch, D., D´ıaz, M. C., & Mangeney, A., A new Savage-Hutter type model for submarine avalanches and generated tsunami, Journal of Computational Physics, 227(16), 2008, 7720-7754.

[4] Salmon, R., Numerical solution of the two-layer shallow water equations with bottom topography, Journal of Marine Research, 60(4), 2002, 605- 638.

[5] Abgrall, R., & Karni, S., Two-layer shallow water system: a relaxation approach, SIAM Journal on Scientific Computing, 31(3), 2009, 1603-1627.

[6] Bouchut, F., & de Luna, T. M., An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, ESAIM: Mathematical Modelling and Numerical Analysis, 42(4), 2008, 683-698.

[7] Balb´as, J., & Karni, S., A non-oscillatory central scheme for one-dimensional two-layer shallow water flows along channels with varying width, Journal of Scientific Computing, 55(3), 2013, 499-528.

[8] Castro-D´ıaz, M. J., Fern´andez-Nieto, E. D., Gonz´alez-Vida, J. M., & Par´es Madroal, C., Numerical treatment of the loss of hyperbolicity of the two-layer shallow-water system, Journal of Scientific Computing, 48(1), 2011, 16- 40.

[9] Ostapenko, V. V., Complete systems of conservation laws for two-layer shallow water models, Journal of Applied Mechanics and Technical Physics, 40(5), 1999, 796-804.

[10] Kurganov, A., & Petrova, G., Central-upwind schemes for two layer shallow water equations, SIAM Journal on Scientific Computing, 31(3), 2009, 1742- 1773.

[11] D´az, M. C., Fern´andez-Nieto, E. D., Gonz´alez-Vida, J. M., Mangeney, A., & Par´es, C., A high-order finite volume method for nonconservative problems and its application to model submarine avalanches, Integral Methods in Science and Engineering, Vol. 2, 291-101, Birkhuser, Boston, 2010. 

[12] Castro, M. J., Garc´ıa-Rodr´ıguez, J. A., Gonz´alez-Vida, J. M., Mac´ıas, J., Par´es, C., & Vzquez-Cendn, M. E., Numerical simulation of two-layer shallow water flows through channels with irregular geometry, Journal of Computational Physics, 195(1), 2004, 202-235.

[13] Pelanti, M., Bouchut, F., Mangeney, A., & Vilotte, J. P., Numerical modeling of two-phase gravitational granular flows with bottom topography, In Hyperbolic problems: theory, numerics, applications, Springer Berlin, Heidelberg, 2008.

[14] Pelanti, M., Bouchut, F., & Mangeney, A., A Roe-type scheme for two-phase shallow granular flows over variable topography, ESAIM: Mathematical Modelling and Numerical Analysis, 42(5), 2008, 851-885.

[15] Pelanti, M., Bouchut, F., & Mangeney, A., A Riemann solver for single phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers, Journal of Computational Physics, 230(3), 2011, 515-550.

[16] Greenberg, J. M., & Leroux, A. Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM Journal on Numerical Analysis, 33(1), 1996, 1-16.

[17] Hubbard, M. E., & Garcia-Navarro, P., Flux difference splitting and the balancing of source terms and flux gradients, Journal of Computational Physics, 165(1), 2000, 89-125.

[18] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms, ESAIM: Mathematical Modelling and Numerical Analysis, 35(4), 2001, 631-645.

[19] Xu, K., Gas-kinetic theory-based flux splitting method for ideal magnetohydrodynamics, Journal of Computational Physics, 153(2), 1999, 334-352.

[20] Mandal, J. C., & Deshpande, S. M., Kinetic flux vector splitting for Euler equations, Computers & Fluids, 23(2), 1994, 447-478.

[21] Zia, S., & Qamar, S., A kinetic flux-vector splitting method for single-phase and two-phase shallow flows, Computers & Mathematics with Applications, 67(6), 2014, 1271-1288.

[22] Qamar, S., & Mudasser, S., A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics, Computer Physics Communications, 181(6), 2010, 1109-1122.

[23] Deshpande, S. M., A Second-Order Accurate Kinetic-Theory-Based Method for Inviscid Compressible Flows, NASA Langley Tech., 2613, 1986.

[24] Pullin, D. I., Direct simulation methods for compressible inviscid ideal gas flow, Journal of Computational Physics, 34, 1980, 231-244.

[25] Rietz, R. D., One-dimensional Compressible Gas Dynamics Calculations Using the Boltzmann Equation, Journal of Computational Physics, 42, 1981, 108.

[26] Perthame, B., Second-Order Boltzmann Schemes for Compressible Euler Equations in One and Two Space Dimensions, SIAM Journal of Numerical Analysis, 29(1), 1992, 1-19.

[27] Chang, S.C., The method of space time conservation element and solution element-A new approach for solving the Navier Stokes and Euler equations, Journal of Computational Physics, 119, 1995, 295-234.

[28] Chang, S.C., Wang, X.Y., & Chow, C.Y., New developments in the method of space-time conservation element and solution element-Applications to two-dimensional time-marching problems, NASA TM 106758, 1994.

[29] Chang, S.C., Wang, X. Y., & Chow, C. Y., The space-time conservation element and solution element method. A new high resolution and genuinely multidimensional paradigm for solving conservation laws, Journal of Computational Physics, 156, 1999, 89-136.

[30] Izem, N., Seaid, M., & Wakrim, M., A discontinuous Galerkin method for two-layer shallow water equations, Mathematics and Computers in Simulation, 120, 2016, 12-23.