Elementary waves, Riemann problem, Riemann invariants and new conservation laws for the pressure gradient model

2019 ◽  
Vol 134 (5) ◽  
Author(s):  
Mahmoud A. E. Abdelrahman ◽  
G. M. Bahaa
Keyword(s):  
Pressure Gradient ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Riemann Invariants ◽  
Gradient Model ◽  
Elementary Waves

On the Shallow Water Equations

2017 ◽  
Vol 72 (9) ◽  
pp. 873-879 ◽  
Author(s):  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water System ◽  
Diagonal Form ◽  
Riemann Invariants ◽  
Interesting Application ◽  
Nonlinear Conservation Laws ◽  
Shallow Water System ◽  
Elementary Waves

AbstractWe studied the shallow water equations of nonlinear conservation laws. First we studied the parametrisation of nonlinear elementary waves and hence we present the solution to the Riemann problem. We also prove the uniqueness of the Riemann solution. The Riemann invariants are formulated. Moreover we give an interesting application of the Riemann invariants. We present the shallow water system in a diagonal form, which admits the existence of a global smooth solution for these equations. The other application is to introduce new conservation laws for the shallow water equations.

The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity

1995 ◽  
Vol 125 (4) ◽  
pp. 675-699 ◽  
Author(s):  
Michael Shearer ◽  
Yadong Yang
Keyword(s):  
Shock Waves ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Mixed Type ◽  
Vector Fields ◽  
Heteroclinic Orbits ◽  
Cubic Nonlinearity ◽  
System Of Conservation Laws ◽  
Admissible Solutions ◽  
Image Position

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation lawswhere σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.

Sign in / Sign up

Export Citation Format

Share Document