High order maximum-principle-preserving discontinuous galerkin method for convection-diffusion equations

© 2015 Society for Industrial and Applied Mathematics. In this paper, we propose to apply the parametrized maximum-principle-preserving (MPP) flux limiter in [T. Xiong, J.-M. Qiu, and Z. Xu, J. Comput. Phys., 252 (2013), pp. 310-331] to the discontinuous Galerkin (DG) method for solving the convection-diffusion equations. The feasibility of applying the proposed MPP flux limiters is based on the fact that cell averages of the DG solutions are updated in a conservative fashion (by using flux difference). Compared with the earlier approach in preserving the MPP property of DG solutions for convection-diffusion problems [Y. Zhang, X. Zhang, and C.-W. Shu, J. Comput. Phys., 234 (2012), pp. 295-316], we avoid the difficulty of rewriting cell averages as a convex combination of point values in the presence of diffusion terms for DG schemes of higher than second order accuracy. As a result, our proposed parametrized MPP flux limiter can be applied to DG methods of arbitrary high order. We theoretically prove that up to the third order accuracy can be preserved for scalar one-dimensional problems. Numerical evidence is presented to show that the proposed MPP flux limiter method does not adversely affect the desired high order accuracy, nor does it require restrictive time steps. Numerical experiments including those for incompressible flow examples demonstrate the high order accuracy preserving, the MPP performance, and the robustness of the proposed method