A simple predictor-corrector scheme without order reduction for advection-diffusion-reaction problems

TL;DR

This paper introduces a predictor-corrector scheme for advection-diffusion-reaction equations that avoids order reduction and achieves second-order accuracy, improving computational efficiency and accuracy in complex boundary conditions.

Contribution

The paper extends a splitting method to nonlinear advection-reaction problems, constructing a predictor-corrector scheme that maintains second-order convergence without order reduction.

Findings

Scheme achieves second-order convergence under regularity assumptions.

Numerical experiments confirm theoretical accuracy improvements.

Method simplifies computations by using explicit Euler steps.

Abstract

Treating diffusion and advection/reaction separately is an effective strategy for solving semilinear advection-diffusion-reaction equations. However, such an approach is prone to suffer from order reduction, especially in the presence of inhomogeneous Dirichlet boundary conditions. In this paper, we extend an approach of Einkemmer and Ostermann [SIAM J. Sci. Comput. 37, A1577-A1592, 2015] to advection-diffusion-reaction problems, where the advection and reaction terms depend nonlinearly on both the solution and its gradient. Starting from a modified splitting method, we construct a predictor-corrector scheme that avoids order reduction and significantly improves accuracy. The predictor only requires the solution of a linear diffusion equation, while the corrector is simply an explicit Euler step of an advection-reaction equation. Under appropriate regularity assumptions on the exact solution, we rigorously establish second-order convergence for this scheme. Numerical experiments are presented to confirm the theoretical results.