算子分裂有限元方法求解二维Burgers方程

The Operator Splitting Finite Element Method for Two-Dimensional Burgers Equation

摘要: 本文提出了一种求解Burgers方程新的算子分裂有限元方法。该算法采用算子分裂法将Burgers方程分解成纯对流部分和扩散部分：对流方程时间离散采取中心差分格式，空间的离散采用标准的Galerkin有限元法；扩散子方程的时间离散采取向后差分格式，空间的离散仍采用标准的Galerkin有限元法。该方法特点是对流部分特殊的显式处理，对其使用多步法技术从根本上扩大稳定性区域，而且多步格式在选择适当步数的条件下可以呈现出无条件稳定。通过数值实验验证了该算法单步和多步格式的稳定性和收敛性，并对其进行了误差估计。

Abstract: This paper proposes a new operator splitting finite element method for two-dimensional Burgers equation. The new method is used to decompose the Burgers equation into pure convection and diffusion part: the time discretization of the convection equation solved by the central difference scheme, and the space discretization by the standard Galerkin finite element method; the time discretization of the diffusion equation solved by the backward difference scheme, and the space discretization still using the standard Galerkin finite element method. The characteristic of this method is that the convection part is specially processed, using multi-step technology to expand the stability of the region and selecting the appropriate number of steps, the multi-step scheme can present unconditionally stable. The stability and convergence of the algorithm are verified by numerical experiments.