Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type

1986 ◽  
Vol 93 (1) ◽  
pp. 45-59 ◽  
Author(s):  
Michael Shearer
Keyword(s):  
Conservation Laws ◽  
Initial Value Problems ◽  
Mixed Type ◽  
Initial Value ◽  
System Of Conservation Laws ◽  
Admissible Solutions

A difference scheme for a stiff system of conservation laws

1998 ◽  
Vol 128 (6) ◽  
pp. 1403-1414 ◽  
Author(s):  
Wen-An Yong
Keyword(s):  
Relaxation Time ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Difference Method ◽  
Initial Value Problems ◽  
Equilibrium System ◽  
Initial Value ◽  
Stiff System ◽  
System Of Conservation Laws ◽  
Entropy Satisfying

A stiff system of conservation laws is analysed using a difference method. The existence of entropy-satisfying BV-solutions to the initial value problems is established. Furthermore, we show that the solutions converge to the solutions of the corresponding equilibrium system as the relaxation time tends to zero.

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.

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