Type: Article
Publication Date: 2020-08-05
Citations: 3
DOI: https://doi.org/10.1515/cmam-2020-0009
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:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {m\geq 1} intervals of length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mi>k</m:mi> <m:mi>m</m:mi> </m:mfrac> </m:math> {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo></m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo></m:mo> <m:mi>k</m:mi> </m:mrow> </m:mrow> </m:math> {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mfrac> <m:msup> <m:mi>C</m:mi> <m:mo>′′</m:mo> </m:msup> <m:mi>m</m:mi> </m:mfrac> </m:mrow> </m:mrow> </m:math> {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.