Abstract We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k m {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form C 0 ⁢ h 2 + C m ⁢ k {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where C m ≤ C ′ + C ′′ m {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}} . This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

中文翻译：

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对流扩散问题的一阶显隐分裂方法

摘要 我们分析了空间周期对流扩散问题的空间二阶、时间一阶精确有限差分方法。该方法是一种基于空间半离散解的一阶李分裂的时间步长方法。在每个时间步长中，在该解的长度为 k 的区间上，该方法对扩散部分使用后向欧拉方法，然后在长度为 m ≥ 1 {m\geq 1} 的区间上应用稳定的显式前向欧拉近似km {\frac{k}{m}} 用于对流部分。h 是空间中的网格宽度，这会导致形式为 C 0 ⁢ h 2 + C m ⁢ k {C_{0}h^{2}+C_{m}k} 的误差界限，以获得适当的平滑解，其中C m ≤ C ′ + C ′′ m {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}} 。这项工作补充了早期的研究 [V. Thomée 和 A.S. Vasudeva Murthy，对流扩散问题的显隐分裂方法，计算。方法应用 数学。19 2019, 2, 283–293] 基于二阶 Strang 分裂。