  Currently, nearly all positivity preserving discontinuous Galerkin (DG)
discretizations of partial differential equations are coupled with explicit
time integration methods. Unfortunately, for many problems this can result in
severe time-step restrictions. The techniques used to develop explicit
positivity preserving DG discretizations can, however, not easily be combined
with implicit time integration methods. In this paper we therefore present a
new approach. Using Lagrange multipliers the conditions imposed by the
positivity preserving limiters are directly coupled to a DG discretization
combined with a Diagonally Implicit Runge-Kutta time integration method. The
positivity preserving DG discretization is then reformulated as a
Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in
constrained optimization. Since the limiter is only active in areas where
positivity must be enforced it does not affect the higher order DG
discretization elsewhere. The resulting non-smooth nonlinear algebraic
equations have, however, a different structure compared to most constrained
optimization problems. We therefore develop an efficient active set semi-smooth
Newton method that is suitable for the KKT formulation of time-implicit
positivity preserving DG discretizations. Convergence of this semi-smooth
Newton method is proven using a specially designed quasi-directional derivative
of the time-implicit positivity preserving DG discretization. The time-implicit
positivity preserving DG discretization is demonstrated for several nonlinear
scalar conservation laws, which include the advection, Burgers, Allen-Cahn,
Barenblatt, and Buckley-Leverett equations.
