Interaction of Shallow Water Waves

In this paper, we consider the Riemann problem and interaction of elementary waves for a nonlinear hyperbolic system of conservation laws that arises in shallow water theory. This class of equations includes as a special case the equations of classical shallow water equations. We study the bore and dilatation waves and their properties, and show the existence and uniqueness of the solution to the Riemann problem. Towards the end, we discuss numerical results for different initial data along with all possible interactions of elementary waves. It is noticed that in contrast to the p-system, the Riemann problem is solvable for arbitrary initial data, and its solution does not contain vacuum state.

[1]  Blake Temple,et al.  Global solutions of the relativistic Euler equations , 1993 .

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

[3]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[4]  Eleuterio F. Toro,et al.  A fast riemann solver with constant covolume applied to the random choice method , 1987 .

[5]  G. B. Whitham,et al.  Motion of a bore over a sloping beach , 1960, Journal of Fluid Mechanics.

[6]  A. Jeffrey,et al.  An Approximate Analytical Method for Describing the Kinematics of a Bore Over a Sloping Beach , 2002 .

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

[8]  Tai-Ping Liu.  Admissible solutions of hyperbolic conservation laws , 1981 .

[9]  J. Johnson,et al.  Global solutions for an extended class of hyperbolic systems of conservation laws , 1969 .

[10]  J. A. Smoller,et al.  On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems. , 1969 .

[11]  K. Karelsky,et al.  Particular solutions and Riemann problem for modified shallow water equations , 2006 .

[12]  J. Chen.  CONSERVATION LAWS FOR THE RELATIVISTIC P-SYSTEM , 1995 .

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

[14]  Alexandre J. Chorin,et al.  Random choice solution of hyperbolic systems , 1976 .

[15]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .