Adapted Lie splitting method for convection-diffusion problems with singular convective term

TL;DR

This paper introduces an Adapted Lie splitting method for convection-diffusion problems with unbounded convective terms, ensuring stability and convergence under certain conditions, supported by numerical experiments.

Contribution

The paper proposes a novel Adapted Lie splitting method that overcomes stability issues in convection-diffusion problems with unbounded convective terms.

Findings

The method converges to first-order under the analytic semigroup framework.

Numerical experiments confirm the stability and effectiveness of the proposed approach.

Abstract

Splitting methods are a widely used numerical scheme for solving convection-diffusion problems. However, they may lose stability in some situations, particularly when applied to convection-diffusion problems in the presence of an unbounded convective term. In this paper, we propose a new splitting method, called the "Adapted Lie splitting method", which successfully overcomes the observed instability in certain cases. Assuming that the unbounded coefficient belongs to a suitable Lorentz space, we show that the adapted Lie splitting converges to first-order under the analytic semigroup framework. Furthermore, we provide numerical experiments to illustrate our newly proposed splitting approach.