Analysis of average bound preserving time-implicit discretizations for convection-diffusion-reaction equation

Abstract

We propose a high-order average bound-preserving limiter for implicit backward differentiation formula (BDF) and local discontinuous Galerkin (LDG) discretizations applied to convection-diffusion-reaction equations. Our approach first imposes cell average bounds of the numerical solution using the Karush-Kuhn-Tucker (KKT) limiter and then enforces pointwise bounds with an explicit bound-preserving limiter. This method reduces the number of constraints compared to using only the KKT system to directly ensure pointwise bounds, resulting in a relatively small system of nonlinear equations to solve at each time step. We prove the unique solvability of the proposed average bound-preserving BDF-LDG discretizations. Furthermore, we establish the stability and optimal error estimates for the second-order average bound-preserving BDF2-LDG discretization. The unique solvability and stability are derived by transforming the KKT-limited cell average bounds-preserving LDG discretizations into a variational inequality. The error estimates are derived using the cell average bounds-preserving inequality constraints. Numerical results are presented to validate the accuracy and effectiveness of the proposed method in preserving the bounds.