  We develop entropy dissipative higher order accurate local discontinuous
Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge-Kutta
(DIRK) methods for nonlinear degenerate parabolic equations with a gradient
flow structure. Using the simple alternating numerical flux, we construct
DIRK-LDG discretizations that combine the advantages of higher order accuracy,
entropy dissipation and proper long-time behavior. The implicit time-discrete
methods greatly alleviate the time-step restrictions needed for the stability
of the numerical discretizations. Also, the larger time step significantly
improves computational efficiency. We theoretically prove the unconditional
entropy dissipation of the implicit Euler-LDG discretization. Next, in order to
ensure the positivity of the numerical solution, we use the Karush-Kuhn-Tucker
(KKT) limiter, which couples the positivity inequality constraint with higher
order accurate DIRK-LDG discretizations using Lagrange multipliers. In
addition, mass conservation of the positivity-limited solution is ensured by
imposing a mass conservation equality constraint to the KKT equations. The
unique solvability and unconditional entropy dissipation for an implicit first
order accurate in time, but higher order accurate in space, KKT-LDG
discretizations are proved, which provides a first theoretical analysis of the
KKT limiter. Finally, numerical results demonstrate the higher order accuracy
and entropy dissipation of the KKT-DIRK-LDG discretizations for problems
requiring a positivity limiter.
