Help With Understanding Riemann Problem for Traffic Flow Modeling

[tanderse]

I'm having a lot of trouble getting my head around this topic.. I am currently trying to create a numerical model of traffic flow, which has worked out to be a nonlinear hyperbolic PDE. If anyone could explain the concept of the Riemann problem to me in lamen terms, it would be greatly appreciated.. I understand it cannot be explained in 2 sentences, but any help would be appreciated.

I guess to be more specific, I do not understand how to use the shock and rarefraction wave analyses to define the boundary states (of each cell as I iterate numerically) from the jump initial conditions. You can assume that the shock and rarefraction wave analyses have been done for me, but I do not understand what they mean or how to implement them. I also do not have a firm grasp on the concept of jump initial conditions.

Any help would be greatly appreciated. thanks

 

[gtoguy97]

.The Riemann problem is a type of initial-value problem for the equations of fluid dynamics. It is named after German mathematician Bernhard Riemann, who first studied the problem in the 19th century. The Riemann problem consists of specifying the initial values for the regions between two discontinuities in a system of equations. The discontinuities represent the boundaries between two different mediums, such as air and water, or two different types of gas. To solve the Riemann problem, the initial conditions are used to identify the type of wave (shock or rarefaction) that will travel across the discontinuities. In order to solve the Riemann problem numerically, one must begin by defining the jump initial conditions. This involves specifying the initial values of the variables on either side of the discontinuity. From these initial conditions, the shock and rarefraction wave analyses can be used to determine the boundary states at each cell. For instance, the shock wave analysis can be used to calculate the speed of the propagating shock wave, while the rarefraction wave analysis can be used to calculate the velocity and pressure of the rarefraction wave. Once the boundary states have been established, the numerical model can be iterated to calculate the solution for the entire domain.