An iterative algorithm for POD basis adaptation in solving parametric convection–diffusion equations

Abstract

In this paper, we develop an iterative algorithm for proper orthogonal decomposition (POD) basis adaptation in solving linear parametric PDEs. Specifically, we consider the convection–diffusion equations with diffusivity as a parameter. To construct POD basis functions for the convection–diffusion equation with a small diffusivity, we need a fine-grid solver to obtain accurate solution snapshots, which leads to a large amount of computation and memory costs. Meanwhile, a coarse-grid solver is sufficient for obtaining accurate solution snapshots for the convection–diffusion equation with a large diffusivity. We aim to adapt the POD basis functions extracted from the solution snapshots of a large diffusivity for the construction of a reduced-order model at a small diffusivity without resorting to a fine-grid solver. Our POD basis adaptation method exploits the implicit dependence of solutions on diffusivity. The POD basis functions are adapted through an iterative algorithm, where the full-order model simulation at a large diffusivity and the POD-based reduced-order model simulation at a small diffusivity are implemented, alternatively. We also provide convergence analysis for our POD basis adaptation method. The algorithm and convergence analysis can be generalized to other types of linear parametric PDEs without any difficulty. Finally, we present numerical results to demonstrate the performance and accuracy of the proposed method.