Time-dependent convection-diffusion problems is considered, particularly when the diffusivity is very small and sharp layers exist in the solutions. Nonphysical oscillations may occur in the numerical solutions when using regular mesh with standard computational methods. In this work, we develop a moving mesh SUPG (MM-SUPG) method, which integrates the streamline upwind Petrov-Galerkin (SUPG) method with the moving mesh partial differential equation (MMPDE) approach. The proposed method is designed to handle both isotropic and anisotropic diffusivity tensors. For the isotropic case, we focus on improving the stability of the numerical solution by utilizing both artificial diffusion from SUPG and mesh adaptation from MMPDE. And for the anisotropic case, we focus on the positivity of the numerical solution. We introduce a weighted diffusion tensor and develop a new metric tensor to control the mesh movement. We also develop conditions for time step size so that the numerical solution satisfies the discrete maximum principle (DMP). Numerical results demonstrate that the proposed MM-SUPG method provides results better than SUPG with fixed mesh or moving mesh without SUPG.

翻译：考虑含时对流扩散问题，特别关注扩散系数极小且解中存在尖锐边界层的情形。当采用常规网格和标准计算方法时，数值解中可能出现非物理振荡。本文发展了一种移动网格SUPG（MM-SUPG）方法，将流线上迎风Petrov-Galerkin（SUPG）方法与移动网格偏微分方程（MMPDE）方法相结合。所提方法旨在处理各向同性和各向异性扩散张量。对于各向同性情形，我们通过结合SUPG的人工扩散与MMPDE的网格自适应，重点提升数值解的稳定性；对于各向异性情形，则聚焦于数值解的正性。我们引入加权扩散张量并开发新度量张量以控制网格运动，同时建立时间步长条件，确保数值解满足离散极大值原理（DMP）。数值结果表明，所提MM-SUPG方法优于固定网格下的SUPG方法或未结合SUPG的移动网格方法。