An initial-boundary corrected splitting method for diffusion-reaction problems
Thi Tam Dang, Lukas Einkemmer, Alexander Ostermann
TL;DR
This work addresses order reduction in Strang splitting for diffusion-reaction problems with nontrivial boundary conditions by introducing an initial-boundary corrected (IBC) splitting that eliminates the need for precomputed boundary corrections. By transforming the problem with a zero-initial-data, homogeneous-boundary correction $z_n(t)$ and splitting the resulting system into diffusion and nonlinear reaction components, the authors prove second-order convergence under analytic semigroups and validate the theory with 1D and 2D numerical experiments. The key contribution is a practical, robust splitting scheme that maintains $O(\tau^2)$ accuracy across Dirichlet, Neumann, Robin, and mixed BC without costly boundary corrections, improving accuracy and efficiency in diffusion-reaction simulations. This advances numerical methods for parabolic problems with oblique boundary conditions and broadens applicability to realistic physical models, including combustion-related scenarios.
Abstract
Strang splitting is a widely used second-order method for solving diffusion-reaction problems. However, its convergence order is often reduced to order $1$ for Dirichlet boundary conditions and to order $1.5$ for Neumann and Robin boundary conditions, leading to lower accuracy and reduced efficiency. In this paper, we consider a new splitting approach, called an initial-boundary corrected splitting, which avoids order reduction while improving computational efficiency for a wider range of applications. In contrast to the corrections proposed in the literature, it does not require the computation of correction terms that depend on the boundary conditions and boundary data. Through rigorous analytical convergence analysis and numerical experiments, we demonstrate the improved accuracy and performance of the proposed method.