# Error estimates for a POD method for solving viscous G-equations in incompressible cellular flows

## TL;DR

This paper introduces a Galerkin POD-based model reduction technique for viscous G-equations in cellular flows, providing error estimates, convergence analysis, and demonstrating improved computational efficiency and accuracy in turbulent flame speed simulations.

## Contribution

The paper develops a novel POD-based model reduction method with rigorous error estimates for viscous G-equations in cellular flows, including convergence analysis and numerical validation.

## Key findings

- The method achieves accurate solutions with reduced computational cost.
- Numerical results confirm the efficiency and accuracy of the POD approach.
- The approach effectively captures turbulent flame speeds in cellular flows.

## Abstract

The G-equation is a well-known model for studying front propagation in turbulent combustion. In this paper, we develop an efficient model reduction method for computing \textcolor{black}{regular solutions} of viscous G-equations in incompressible steady and time-periodic cellular flows. Our method is based on the Galerkin proper orthogonal decomposition (POD) method. To facilitate the algorithm design and convergence analysis, we decompose the solution of the viscous G-equation into a mean-free part and a mean part, where their evolution equations can be derived accordingly. We construct the POD basis from the solution snapshots of the mean-free part. With the POD basis, we can efficiently solve the evolution equation for the mean-free part of the solution to the viscous G-equation. After we get the mean-free part of the solution, the mean of the solution can be recovered. We also provide rigorous convergence analysis for our method. Numerical results for \textcolor{black}{viscous G-equations and curvature G-equations} are presented to demonstrate the accuracy and efficiency of the proposed method. In addition, we study the turbulent flame speeds of the viscous G-equations in incompressible cellular flows.