The Riemann Problem And Interactions Of Elementary Waves For Some Non-conservative Hyperbolic Equations

This article mainly concerned the Riemann problem and the interaction of elementary waves of non-conservative hyperbolic equations,which includes:isentropic flow in a vari-able cross-section duct,one dimensional blood flow model in arteries,two-phase flow model in a variable cross-section area and a traffic flow model on a road with variable widths.In Chapter 1,we introduce the background and some development of compressible equations,especially the hyperbolic Euler equations.In Chapter 2,we introduce some concepts of hyperbolic system.For example,the definition of simple waves and weak solution.We mainly deduce the equations of non-isentropic flow in a variable cross-section area based on the conservation laws.In Chapter 3,we give characteristic analysis and elementary waves of isentropic flow in a variable cross-section area.We recall the Riemann problem of duct flow briefly,and then use the characteristic analysis method to discuss the interaction of elementary waves,especially the interaction of rarefaction wave or shock wave with stationary wave.We also show the behavior of the solutions in large time scale based on the p-system.In Chapter 4,we mainly consider the Riemann problem of one dimensional blood flow model in arteries.To avoid the non-uniqueness of the solutions,we introduce the global entropy condition on the stiffness coefficient of the vessel.We construct the Riemann problem in the whole phase plane.According to the Riemann problem,we know that the solution loss uniqueness under some initial data.To select the physical solution,we first define the energy function of blood flow model,then we claim that the physical solution is one that maximize the energy function.This conclusion is verified by some numerical results in the end of this chapter.In Chapter 5,we first introduce the two-phase flow model,then we establish a two-phase flow model in a variable cross-section duct.After that,we give some characteristic analysis of the new model and obtain the elementary waves of two-phase flow in a variable cross-section area.To ensure the uniqueness of the Riemann problem,we introduce the global entropy conditions on both the cross-section area and the volume fraction.Some numerical results are given in this chapter.In Chapter 6,we mainly concerned the traffic flow model on a road with variable widths.One of the differences between traffic flow model and compressible flow lies on the "pseudo pressure p=p(p)" in the traffic model.We use the "pseudo pressure" to denote the velocity offset and then deduce the traffic flow model on a road with variable widths.This new model is non-conservative and global entropy condition is needed to construct the Riemann solution.By solving the Riemann solution,we conclude that the solution is unique when the widths of the road increase,the same conclusion is not true when the widths of the road decrease.