An initial-corrected splitting approach for convection-diffusion-reaction problems
Thi Tam Dang, Lukas Einkemmer, Alexander Ostermann
TL;DR
The paper tackles order reduction observed in Strang splitting for convection-diffusion-reaction problems with Dirichlet boundaries. It introduces an initial-corrected splitting that subtracts the step’s initial data to enforce homogeneous Dirichlet conditions, then applies a Strang-type splitting to the transformed system; for time-dependent boundaries, a first-order correction $z_n(t)$ is used. The authors prove a global second-order convergence bound $∥u_n-u(t_n)∥ ≤ C τ^2 (1 + |log τ|)$ under standard analytic-semigroup and Lipschitz assumptions, and validate the theory with numerical experiments in 1D and 2D showing robust second-order accuracy across boundary condition types. Overall, the method provides a simple, efficient, and reliable way to avoid order reduction in practical convection-diffusion-reaction simulations with Dirichlet BCs.
Abstract
Splitting methods constitute a well-established class of numerical schemes for solving convection-diffusion-reaction problems. They have been shown to be effective in solving problems with periodic boundary conditions. However, in the case of Dirichlet boundary conditions, order reduction has been observed even with homogeneous boundary conditions. In this paper, we propose a novel splitting approach, the so-called `initial-corrected splitting method', which succeeds in overcoming order reduction. A convergence analysis is performed to demonstrate second-order convergence of this modified Strang splitting method. Furthermore, we conduct numerical experiments to illustrate the performance of the newly developed splitting approach.