  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.
