Solving the advection equation with artificial damping I'm having trouble with the following system of partial differential equations:
$$
\frac{\partial u_1}{\partial t} + a \frac{\partial u_1}{\partial x}=u_2(x,t)\;, \\
\frac{\partial u_2}{\partial t} +  b u_2(x,t) = ba \frac{\partial u_1}{\partial x}\;, 
$$
where $a$ and $b$ are real positive constants. 
If $b=0$, this reduces to the advection equation, so with the initial condition $u_1(x,0)=f(x)$ the solution is a right-running wave $u_1(x,t)=f(x-at)$. 
This (simplified) system may be used in a finite-difference model with $b>0$ to artificially damp such a wave over time to simulate an infinite medium.  
However, I noticed that with $b>0$, $u_1(x,0)=f(x)$ and $u_2(x,0)=0$ the solution for $u_1$ does not depend on time and becomes $u_1(x,t)\simeq f(x)$. 


*

*Would anyone be able to explain this?

*More specifically, I would like to find the initial condition $u_2(x,0)=g(f(x))$ so that $u_1(x,t)=0$ as $t\rightarrow\infty$.


Thanks!

For arbitrary differentiable functions $f$ and $g$ you have solutions
$$ \eqalign{u_1(x,t) &= f(x) + \exp(-bx/a) g(x-at)\cr
            u_2(x,t) &= a f'(x) - b \exp(-bx/a) g(x-at)\cr} $$
For arbitrary continuous initial conditions $u_1(x,0)$, $u_2(x,0)$ you can take 
$$g(x) = \exp(bx/a) (-f(x) + u_1(x,0))$$
where $f(x)$ is a solution of the ODE
$$ a f'(x) + b f(x) = u_2(x,0) + b u_1(x,0) $$
namely
$$ f(x) = \frac{\exp(-bx/a)}{a} \left( \int (u_2(x,0)+b u_1(x,0)) \exp(bx/a)\; dx + c\right) $$
