The matrix A in the system of Euler Equations I am running a simulation on a 1D Euler equations:
$$\frac{\partial \rho}{\partial t}+ \frac{\partial (\rho u)}{\partial x}=0$$
$$\frac{\partial (\rho u)}{\partial t}+ \frac{\partial (\rho u^2+\rho)}{\partial x}=0$$$$\frac{\partial E}{\partial t}+ \frac{\partial ((E+P)u)}{\partial x}=0$$
The above system can be written into a vector equation:
$$\frac{\partial U}{\partial t}+ \frac{\partial (F(U))}{\partial x}=0$$, where $U=(\rho,\rho u, E)^T$, $F(U)=(\rho u, \rho u^2, (E+P)u)^T$. In addition, to make the system closed, the following relations are imposed:
$$E=\frac{1}{2}\rho u^2+\rho e, \ e=\frac{P}{(\gamma-1)\rho}$$
I am using Godunov's 2nd order method. Owing to the complexity of dealing with a nonlinear system, I have to convert the vector equations into something like this:
$$\frac{\partial U}{\partial t}+ A\frac{\partial U}{\partial x}=0$$, where $A$ is a matrix. $A$ is not necessarily a constant matrix in general setting, but locally it is. Can someone help me in determining what $A$ is? 
Thank you.

A is the jacobian matrix of the system.
basically it is partial derivative of variable with respect to independent variables.
http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
It is used to decouple the coupled equations,so that solving can be easily done.
Im too working on euler equations but no luck still.Im coding in modelica.
