I'm currently trying to get into this topic.  I've learned that the basic scheme for the advection problem ($D_{x}u+a*D_{t}u=0$) can be solved in a scheme like $$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}au^{k}_{h}=-(au_{h})^{*}\psi(x^{k}_{r})+(au_{h})^{*}\psi(x^{k}_{l})$$Here, we have $$ M^{k}_{ij}=(\psi_{i},\psi_{j}),  S^{k}_{ij}=(\psi_{i},\frac{d\psi_{j}}{dx})$$ $\psi_{i}$ are the basis functions. I think I understand this scheme but I'm not quite how to deal with the case of an inhomogeneous PDE.  How has the scheme to be modified if I've got $f(x,t)$ instead of 0 above?
Discontinuos Galerkin Method - inhomogeneous advection problem
pde;boundary conditions;discontinuous galerkin;advection