Let $C = 1$ and $\gamma = 1$. and the initial conditions as $U(t=0) = e^{cos(2\pi x)}$ (periodic boundary) and $V(t=0) = 0$. Now I'm getting two numerical solutions. One models the $U$ as a transported wave moving to the right, the other models $U$ as a decaying wave over time. Which one is right?
$$\frac{\partial U}{\partial t}+C\frac{\partial U}{\partial x}=0$$ There is no difficulty to analytically solve this PDE thanks to the method of characteristics or other method. The general solution is : $$U(t,x)=F(x-Ct)$$ where $F$ is an arbitrary function.
Initial condition : $U(0,x)=e^{\cos(2\pi x)}=F(x-0t)$ $$F(x)=e^{\cos(2\pi x)}$$ Now the function $F$ is known. We put it into the above general solution where the variable is not $x$ but is $(x-Ct)$. $$\boxed{U(x,t)=e^{\cos\left(2\pi (x-Ct)\right)}}$$ So the exact model for $U(t,x)$ is determined and this answers to your question.