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.

翻译：分裂方法是求解对流-扩散问题中广泛使用的数值格式。然而，在某些情况下，特别是当应用于存在无界对流项的对流-扩散问题时，这些方法可能会失去稳定性。本文提出了一种新的分裂方法，称为"自适应Lie分裂方法"，该方法成功克服了某些情形下观察到的数值不稳定性。在假设无界系数属于适当Lorentz空间的条件下，我们证明了在解析半群框架下自适应Lie分裂方法具有一阶收敛性。此外，我们通过数值实验验证了所提出的新型分裂方法的有效性。