about entropy stability
 

hi could some one tell me how the following identity on entropy stability for a hyperbolic conservation law was derived.. could u give physical relevance too?

U(u)_t + F(U)_x < 0

Here U is a entropy function and F an entropy flux function

and also tell me how this condition gives a relevant weak solution for a problem. also tell me why do we get more than one weak solutions for a hyperbolic problem.

thank you

Re: about entropy stability
 

Actually, there are other entropy conditions such as Lax and Oleinik conditions. They are more useful in practical code design because they put restrictions on wave speed. Entropy fix in Flux Difference Scheme is one example of Lax entropy condition.

As far as your question regarding the entropy function and entropy flux function goes, this relation was derived from the observation that the entropy function is conserved when the solution is smooth (equality sign applies) and it it greater or less than zero at the discontinuity. At the discontinuous point, entropy equation has a source or sink term guarantying that the entropy at the time t2 is either less or greater than the entropy at time instant t1. The sign <= in the equation is chosen from the mathematical requirement that the entropy flux function is convex - nothing magical about it but sometimes source of confusion for students who are used to the idea that the physical entropy is non-decreasing quantity globally.

We get more than one week solution for Euler equations because they do admit non-physical solutions such as expansion shocks. This is just the nature of the mathematical system and has nothing to do with physics. In fact it tells you that Euler equations do not represent physical system correctly and they need additional information in order to yield unique solution. This is achieved by constraining the inviscid solution by special treatment that is based on entropy to produce physical solutions.

Entropy equation is not commonly used to produce numerical schemes that have unique solution. This equation is used more in mathematical analysis of the hyperbolic systems of equations because it fits well into the mathematical framework of conservation laws. Lax and Oleinik conditions are not as much used in theoretical context, they are used to produce practical solutions in practice.

Re: about entropy stability
 

Dear sir,

That was indeed helpful. And i hope it would be better if i get an example problem worked out. take a one dimensional unsteady wave equation.. or burger's equation. i use a distribution function and obtain the weak formulation. now what are the solutions to this formulation? i mean what are the weak solutions? if the boundary conditions are clearly specified, can there still be more than one solution? Because i am confusing it with weak solutions of second degree BYP. How do we determine the number of weak solutions? is there a book or paper that gives an elementary insight? I have my project on this and am running out of time.. thank you

Re: about entropy stability
 

Consider Cauchy 1D problem with smooth initial data. One example of 1D Cauchy problem is simply hyperbolic scalar conservation defined on real axis (no boundaries). Nonlinear flux function has an ability to take initially smooth profile at time t=0 and turn it into discontinuous profile at a later time t. The key here is the nonlinearity of the flux function and initial profile that creates weak solutions. Weak solution is a shock wave, expansion wave and slip discontinuity for ideal gases. Typical example that is rich in weak solutions is easily obtained on the computer by setting up the shock tube problem. Additionally, observe that Cauchy problem has no boundaries and therefore it is not BVP.

As a hint for your problem consider hyperbolic problem described with the following equation given as Latex equation:

\begin{equation}\label{scalar} \frac{\partial u}{\partial t} + \frac{\partial f}{\partial x} = 0 \end{equation}

Associate entropy transport equation with this equation as follows:

\begin{equation}\label{entropy} \frac{\partial U(u)}{\partial t} + \frac{\partial F(u)}{\partial x} \le 0 \end{equation}

It can be shown that equation \ref{scalar} implies equation \ref{entropy} with equality sign away from the shocks if entropy flux function satisfies the following relation:

\begin{equation}\label{condition1} \frac{d U(u)}{d u} = \frac{d f}{d u}\frac{d U}{d u} \end{equation}

Furthermore, weak solutions of \ref{scalar} satisfy \ref{entropy} across shocks with less-than sign if the entropy flux function is convex, i.e.

\begin{equation}\label{condition2}

\frac{d^2 U}{{d u}^2} \ge 0 \end{equation}

I hope this helps and good luck.

Re: about entropy stability
 

Dear sir,

I have myself performed the computation of the shock tube problem earlier and obtained results. From what u said, i understand that the solution i obtained was a weak solution. Now my question is "I have only got one solution. where are the other physically irrelevant solutions. I never used an inequality in my equations." I hope now my question is clear. Its about how do i get the other weak solutions. i am getting only one solution on my computer. How many other would be there?

thanking you.. and sorry for this irritation

prapanj

Re: about entropy stability
 

What is meant by non-unique solution can be seen even on the simple example of Burger's equation. It is rather easy to construct one parameter family of solution that satisfy given initial conditions. Here the parameter can be arbitrarily chosen and it does not appear in equations nor it is physically related to the problem necessarily. This means that any of these solutions can satisfy Burgers equation away from the boundaries and some way of choosing among different solution is required. This is where entropy condition comes into play. Furthermore, entropy condition does not even have to appear explicitly as an inequality, all it has to do is to chose the right solution.

In your case when you do numerical simulation, this condition maybe already provided to you by the numeric flux function or by numerical dissipation. Either way, you will still be using entropy condition, sometimes even without realizing that it is being used.