We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be stable and maximum-principle-preserving (MPP) with no step size restriction. The schemes are based on a two-tiered limiting strategy, starting with a high-order limiter-based method that may have small oscillations or maximum-principle violations, followed by an additional limiting step that removes these violations while preserving high order accuracy. The desirable properties of the resulting schemes are demonstrated through several numerical examples.
Bibliographical noteKAUST Repository Item: Exported on 2022-09-14
Acknowledgements: This work was funded by King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. We are grateful to Prof. Dmitri Kuzmin for important discussions that formed the basis of this work, for providing feedback on drafts of the paper and for suggesting the fixed point iteration (31).
ASJC Scopus subject areas
- Computational Theory and Mathematics
- Theoretical Computer Science