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 cannot, however, 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 nonsmooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semismooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semismooth 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.
- Discontinuous Galerkin methods
- Implicit time integration methods
- Karush-Kuhn-Tucker equations
- Maximum principle
- Positivity preserving
- Semismooth Newton methods