In this talk I will present some recent work joint with Jue Yan on developing local discontinuous Galerkin methods for solving time dependent partial differential equations containing
third spatial derivatives (KdV type equations), fourth spatial  derivatives (time dependent bi-harmonic type equations) and fifth  spatial derivatives.  We will emphasize the role of numerical fluxes as the key for a stable and accurate approximation of these methods.  For these new methods we prove $L^2$ stability for general nonlinear problems.  Preliminary numerical examples are shown to  illustrate these methods.  Finally, we present results on a  post-processing technique, originally designed for methods with  good negative-order error estimates, on the local discontinuous  Galerkin methods applied to equations with higher derivatives.

Numerical experiments show that this technique works as well for  the new higher derivative cases, in effectively doubling the rate  of convergence with negligible additional computational cost, for  linear as well as some nonlinear problems, with a uniform mesh.