In this paper, we propose an augmented subspace based adaptive proper
orthogonal decomposition (POD) method for solving the time dependent partial
differential equations. By augmenting the POD subspace with some auxiliary
modes, we obtain an augmented subspace. We use the difference between the
approximation obtained in this augmented subspace and that obtained in the
original POD subspace to construct an error indicator, by which we obtain a
general framework for augmented subspace based adaptive POD method. We then
provide two strategies to obtain some specific augmented subspaces, the random
vector based augmented subspace and the coarse-grid approximations based
augmented subspace. We apply our new method to two typical 3D
advection-diffusion equations with the advection being the Kolmogorov flow and
the ABC flow. Numerical results show that our method is more efficient than the
existing adaptive POD methods, especially for the advection dominated models.Comment: 28 pages, 4 figures, 7 table