I want to solve the Euler equations in 1D numerically. I am currently using DG-Lax Friedrich flux splitting technique and RK-4 explicit time stepping scheme.
Now,using the idea of characteristics it is possible to decompose the Jacobian matrix into incoming and outgoing waves depending on the sign on the sign of the eigen values. Here I am using the idea of characteristics variables where I shift my problem into a characteristic variable. For eg . $ A = W \Lambda W^{-1}$,Then, $w = W^{-1}u$ ;, where u is the unkown of vectors and w is the characteristic variable.
I can solve the problem using both techniques, but I am struggling to find the equivalence. Where does the idea of characteristics come into picture in the the DG scheme ?