斯托克斯方程周期边界条件下一类高阶IMEX格式的稳定性与误差分析
Stability and error analysis of a class of high-order IMEX schemes for Navier-stokes equations with periodic boundary conditions
摘要：
对具有周期边界条件的不可压斯托克斯方程构造了时间上半离散和空间上全离散（Fourier-Galerkin）的高阶格式，并进行了相应的误差分析.该格式是基于标量辅助变量（SAV）方法的隐-显格式。结果表明，这些格式的数值解是一致有界的，对时间步长没有任何限制。这些统一的界使我们能够以统一的形式对直到五阶的格式进行严格的误差分析，并在二维情况下得到$l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$的全局误差估计，在三维情况下得到$l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$的局部误差估计.我们还提出了数值结果证实了我们的理论收敛速度，并证明了高阶格式的优势，在双剪切层问题的复结构的流动。
英文摘要：
We construct high-order semi-discrete-in-time and fully discrete (with Fourier-Galerkin in space) schemes for the incompressible Navier-Stokes equations with periodic boundary conditions, and carry out corresponding error analysis. The schemes are of implicit-explicit type based on a scalar auxiliary variable (SAV) approach. It is shown that numerical solutions of these schemes are uniformly bounded without any restriction on time step size. These uniform bounds enable us to carry out a rigorous error analysis for the schemes up to fifth-order in a unified form, and derive global error estimates in $l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$ in the two dimensional case as well as local error estimates in $l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$ in the three dimensional case. We also present numerical results confirming our theoretical convergence rates and demonstrating advantages of higher-order schemes for flows with complex structures in the double shear layer problem.
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