Examples of system of one dimensional first order hyperbolic PDEs with source term

Question

I am writing an algorithm to solve a system of one- dimensional first order hyperbolic PDEs with constant coefficient matrices and I require some example problems which are already solved analytically or numerically with initial and boundary conditions.

The system of PDEs is of the form: $$A\dot{U} + BU^{'} + F(U) = 0$$ Where, $A$, and $B$ are constant matrices and $F(U)$ is a linear or nonlinear function of $U$.

$\dot{\begin{Bmatrix} \\ \end{Bmatrix}}$ represents the derivative w.r.t. time ($t$), and $\begin{Bmatrix} \\ \end{Bmatrix}^{'}$ represents the derivative w.r.t. spatial variable ($s$).

Example:

One of such problems is the transmission equation given by:

$$\begin{bmatrix}L & 0\\0 & C\end{bmatrix}\dot{\begin{Bmatrix}I \\V\end{Bmatrix}} + \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\begin{Bmatrix}I \\V\end{Bmatrix}^{'} + \begin{Bmatrix} RI \\GV \end{Bmatrix} = \begin{Bmatrix}0 \\0 \end{Bmatrix}$$

$I$ represents current, and $V$ represents voltage.

Where, $L = 4$, $R = 4$, $C = 1$, and $G = 1$

The initial conditions are given by:

$$I(s, 0) = I_0$$ $$V(s, 0) = V_0$$

An analytical solution is available for the above example problem, when $RC = LG$

If anyone has knowledge of similar system of equations (with 3 or more variables if possible) with initial and boundary conditions please let me know. I have searched online and could not find another example.

Answer 1

The scalar case $$a\dot u + b u' + c u =0$$ has analytical closed-form solutions deduced from the method of characteristics. It's much more difficult to do so in the case of systems $$ A\dot{U} + B U' + F(U) = 0 $$ where $F(U)$ is nonzero. Nevertheless, some boundary-value problems can be solved using the Fourier method in the case where $F(U) = CU$ is a linear function. I'd suggest to analyse the scalar case in Fourier space before moving to other systems.

This type of system represents lossy wave propagation. As for examples, I can suggest the case of the damped wave equation (telegraph equation), of linear viscoelastic waves (Maxwell model), or of linear poroelasticity (Biot equations). You might find some other example in the literature related to this post.