The matrix A in the system of Euler Equations

Question

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.

Answer 1

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.

Answer 2

A is the Jacobian matrix which can be written in terms of both primitive and conservative variables. You already defined the conservative variables (U). To distinguish conservative variables (U) from the speed u, the conservative variables are shown with q such as:

$$ \mathbf{Q} = \left[ \begin{matrix} q_1 \\ q_2 \\ q_3 \end{matrix} \right] = \left[ \begin{matrix} \rho \\ \rho u \\ E \end{matrix} \right] $$

Whereas the primitive variables are:

$$ \mathbf{P} = \left[ \begin{matrix} \rho \\ u \\ E \end{matrix} \right] $$

You have given fluxes (F) in terms of primitive variables. F can also be written in terms of conservative variables such as:

$$ \mathbf{F} = \left[ \begin{matrix} f_1 \\ f_2 \\ f_3 \end{matrix} \right] = \left[ \begin{matrix} q_2 \\ \frac{q_2^2}{q_1} \left( 1-0.5\hat{\gamma} \right) + \hat{\gamma}q_3 \\ \frac{q_2}{q_1} \left( q_3\gamma - \frac{\hat{\gamma}q_2^2}{2q_1} \right) \end{matrix} \right] $$

where, $\gamma$ is the ratio of specific heats and $\hat{\gamma} = \gamma - 1$.

A is defined as:

$$ \mathbf{A} = \frac{\partial \mathbf{F}}{\partial \mathbf{Q}} = \left[ \begin{matrix} \frac{\partial f_1}{\partial q_1} & \frac{\partial f_1}{\partial q_2} & \frac{\partial f_1}{\partial q_3} \\ \frac{\partial f_2}{\partial q_1} & \frac{\partial f_2}{\partial q_3} & \frac{\partial f_2}{\partial q_3} \\ \frac{\partial f_3}{\partial q_1} & \frac{\partial f_3}{\partial q_2} & \frac{\partial f_3}{\partial q_3} \end{matrix} \right] $$

It is left to take derivatives one by one. As a result, A is (I worked out the derivatives myself so it may contain mistakes):

$$ \mathbf{A} = \left[ \begin{matrix} 0 & 1 & 0 \\ -\frac{q_2^2}{q_1^2} \left( 1-0.5\hat{\gamma} \right) & 2\frac{q_2}{q_1} \left( 1-0.5\hat{\gamma} \right) & \hat{\gamma} \\ -\frac{\gamma q_2 q_3}{q_1^2} + \frac{\hat{\gamma}q_2^3}{q_1^3} & \frac{\gamma q_3}{q_1} - 3\frac{\hat{\gamma}q_2^2}{2q_1^2} & \frac{q_2\gamma}{q_1} \end{matrix} \right] $$

A can also be written in terms of primitive variables such as (taken from reference 1 (see below)):

$$ \mathbf{A} = \left[ \begin{matrix} 0 & 1 & 0 \\ \hat{\gamma}H - u^2 - a^2 & (3-\gamma)u & \hat{\gamma} \\ \frac{1}{2}\left[(\gamma - 3)H-a^2\right] & H-\hat{\gamma}u^2 & \gamma u \end{matrix} \right] $$

where, H and a are the total enthalpy and the speed of sound, respectively:

$$H = \frac{E+p}{\rho}$$

$$a = \sqrt{\frac{\gamma p}{\rho}}$$

Reference 1: Riemann Solvers and Numerical Methods for Fluid Dynamic, F. Toro