Solving Advection–Diffusion Equation by Proper Generalized Decomposition with Coordinate Transformation

Inheriting a convergence difficulty explained by the Kolmogorov N-width, the advection–diffusion equation is not effectively solved by the proper generalized decomposition (PGD) method. In this paper, we propose a new strategy: proper generalized decomposition with coordinate transformation (CT-PGD). Converting the mixed hyperbolic-parabolic equation to a parabolic one, it resumes the efficiency of convergence for advection-dominant problems. Combining PGD with CT-PGD, we solve advection–diffusion equation by much fewer degrees of freedom, hence improve the efficiency. The advection-dominant regime and diffusion-dominant regime are quantitatively classified by a threshold, computed numerically. Moreover, we find that appropriate preconditioners may further improve the effectiveness.