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1

There is a lot of one can mention here, so I am going to kick off things. CFL is never a sufficient criterion for stability, but a necessary one. That said, just because you adhere the CFL condition, there is no guarantee that you get stable results. This makes absolute sense, since your methods can be arbitrarily bad. Looks to me you are trying not to ...


2

You cannot specify just a boundary condition on $\partial_x u$ at $x_L$. Remember that in DG your solution is composed of piecewise discontinuous polynomials; at every interface you can have effectively any jump in $u$, which is separate from the slope of u to the left or right of the interface. Unless $f(u,x)$ has any dependence on $\partial_x u$ (in which ...


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