The Riemann problem for the shallow water equations with horizontal temperature gradients

Abstract We consider the Riemann problem for the system of shallow water equations with horizontal temperature gradients (the Ripa system). The model under investigation has the form of a nonconservative system, and it is hyperbolic, but is not strictly hyperbolic. We construct all solutions of the Riemann problem. It turns out that there may be up to three distinct solutions. A resonant phenomenon which causes the colliding shock waves is observed, where multiple waves associated with different characteristic fields propagate with the same shock speed.

[1]  Giorgio Rosatti,et al.  The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations , 2010, J. Comput. Phys..

[2]  Manuel Jesús Castro Díaz,et al.  A HLLC scheme for Ripa model , 2016, Appl. Math. Comput..

[3]  Mai Duc Thanh,et al.  The Riemann Problem for a Nonisentropic Fluid in a Nozzle with Discontinuous Cross-Sectional Area , 2009, SIAM J. Appl. Math..

[4]  Philippe G. LeFloch,et al.  A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime , 2011, J. Comput. Phys..

[5]  G. Warnecke,et al.  The Riemann problem for the Baer-Nunziato two-phase flow model , 2004 .

[6]  Blake Temple,et al.  Convergence of the 2×2 Godunov Method for a General Resonant Nonlinear Balance Law , 1995, SIAM J. Appl. Math..

[7]  D. Stewart,et al.  Two-phase modeling of deflagration-to-detonation transition in granular materials: A critical examination of modeling issues , 1999 .

[8]  A. Tzavaras,et al.  Representation of weak limits and definition of nonconservative products , 1999 .

[9]  P. Ripa.  On improving a one-layer ocean model with thermodynamics , 1995, Journal of Fluid Mechanics.

[10]  C. Chalons,et al.  Relaxation and numerical approximation of a two-fluid two-pressure diphasic model , 2009 .

[11]  Mai Duc Thanh,et al.  Numerical Solutions to Compressible Flows in a Nozzle with Variable Cross-section , 2005, SIAM J. Numer. Anal..

[12]  M. Thanh.  A phase decomposition approach and the Riemann problem for a model of two-phase flows , 2014 .

[13]  Yu Liu,et al.  Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients , 2014, Numerische Mathematik.

[14]  M. Thanh,et al.  The Riemann Problem for Fluid Flows in a Nozzle with Discontinuous Cross-Section , 2003 .

[15]  D. H. Cuong,et al.  Building a Godunov-type numerical scheme for a model of two-phase flows , 2017 .

[16]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .

[17]  M. Thanh,et al.  The minimum entropy principle for fluid flows in a nozzle with discontinuous cross-section , 2007, 0712.3774.

[18]  Mai Duc Thanh,et al.  Constructing a Godunov-type scheme for the model of a general fluid flow in a nozzle with variable cross-section , 2017, Appl. Math. Comput..

[19]  P. Goatin,et al.  The Riemann problem for a class of resonant hyperbolic systems of balance laws , 2004 .

[20]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[21]  P. Raviart,et al.  A Godunov-type method for the seven-equation model of compressible two-phase flow , 2012 .

[22]  Donald W. Schwendeman,et al.  The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow , 2006, J. Comput. Phys..

[23]  Mai Duc Thanh,et al.  A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section , 2015, Appl. Math. Comput..

[24]  Gerald Warnecke,et al.  On the Solution to The Riemann Problem for the Compressible Duct Flow , 2004, SIAM J. Appl. Math..

[25]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[26]  E. Isaacson,et al.  Nonlinear resonance in systems of conservation laws , 1992 .

[27]  P. Floch.  Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form , 1989 .

[28]  Ramaz Botchorishvili,et al.  Equilibrium schemes for scalar conservation laws with stiff sources , 2003, Math. Comput..

[29]  C. Klingenberg,et al.  Well-balanced central finite volume methods for the Ripa system , 2015 .

[30]  M. Thanh,et al.  The Riemann problem for the shallow water equations with discontinuous topography , 2007, 0712.3778.

[31]  P. Ripa.  Conservation laws for primitive equations models with inhomogeneous layers , 1993 .