# Strang splitting method for semilinear parabolic problems with inhomogeneous boundary conditions: a correction based on the flow of the nonlinearity

## TL;DR

This paper introduces a new correction method for the Strang splitting technique applied to semilinear parabolic problems with inhomogeneous boundary conditions, reducing computational effort and improving performance especially for stiff nonlinearities.

## Contribution

A novel correction based on the flow of the nonlinearity that requires only one evaluation per step, simplifying implementation and enhancing efficiency for stiff problems.

## Key findings

- The new correction reduces computational effort compared to previous methods.
- Numerical experiments show improved performance with stiff nonlinearities.
- The method achieves convergence for smooth nonlinearities.

## Abstract

The Strang splitting method, formally of order two, can suffer from order reduction when applied to semilinear parabolic problems with inhomogeneous boundary conditions. The recent work [L .Einkemmer and A. Ostermann. Overcoming order reduction in diffusion-reaction splitting. Part 1. Dirichlet boundary conditions. SIAM J. Sci. Comput., 37, 2015. Part 2: Oblique boundary conditions, SIAM J. Sci. Comput., 38, 2016] introduces a modification of the method to avoid the reduction of order based on the nonlinearity. In this paper we introduce a new correction constructed directly from the flow of the nonlinearity and which requires no evaluation of the source term or its derivatives. The goal is twofold. One, this new modification requires only one evaluation of the diffusion flow and one evaluation of the source term flow at each step of the algorithm and it reduces the computational effort to construct the correction. Second, numerical experiments suggest it is well suited in the case where the nonlinearity is stiff. We provide a convergence analysis of the method for a smooth nonlinearity and perform numerical experiments to illustrate the performances of the new approach.