# Two-Grid based Adaptive Proper Orthogonal Decomposition Algorithm for Time Dependent Partial Differential Equations

## TL;DR

This paper introduces a two-grid adaptive POD algorithm for efficiently solving time-dependent PDEs, using error indicators from coarse grids to improve fine grid solutions, demonstrating superior efficiency in numerical experiments.

## Contribution

The paper presents a novel two-grid adaptive POD method that enhances efficiency and accuracy in solving time-dependent PDEs compared to existing POD approaches.

## Key findings

- The method reduces computational cost compared to traditional POD methods.
- Numerical results show improved accuracy in advection-diffusion problems.
- The approach is easy to implement and adaptable to different flow scenarios.

## Abstract

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new algorithm is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection-diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.