An efficient model reduction method for solving viscous G-equations in incompressible cellular flows

The G-equation is a well-known model for studying front propagation in turbulent combustion. In this paper, we shall develop an efficient model reduction method for solving viscous G-equations in incompressible steady and time-periodic cellular flows. Our method is based on the Galerkin proper orthogonal decomposition (POD) methods. 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 numerical method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method. Specifically, we study the turbulent flame speeds of the viscous G-equations in incompressible cellular flows based on the POD method and fully resolved computations.