Positivity-preserving discontinuous Galerkin (DG) methods for solving
hyperbolic conservation laws have been extensively studied in the last
several years. But nearly all the developed schemes
are coupled with explicit time discretizations. Explicit
discretizations suffer from the constraint for the Courant-Friedrichs-Levis (CFL)
number. This makes explicit methods impractical for problems involving
unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes
are often popular in practice, especially in the computational fluid dynamics
(CFD) community. In this paper we develop a high-order positivity-preserving
DG method with the backward Euler time discretization for
conservation laws. We focus on one spatial dimension, however
the result easily generalizes to
multidimensional tensor product meshes and polynomial
spaces.
This work is based on a generalization of the positivity-preserving limiters in
(X. Zhang and C.-W. Shu, Journal of Computational Physics, 229
(2010), pp.~3091--3120) and (X. Zhang and C.-W. Shu, Journal of
Computational Physics, 229 (2010), pp.~8918--8934) to implicit time
discretizations. Both the analysis and numerical experiments indicate that
a lower bound for the CFL number is required to obtain the
positivity-preserving property. The proposed scheme not only preserves the
positivity of the numerical approximation without compromising the designed
high-order accuracy, but also helps accelerate the convergence towards the
steady-state solution and add robustness to the nonlinear solver. Numerical
experiments are provided to support these conclusions.