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

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