Mode multigrid - A novel convergence acceleration method

Abstract

This paper proposes a mode multigrid (MMG) method, and applies it to accelerate the convergence of the steady state flow on unstructured grids. The dynamic mode decomposition (DMD) technique is used to analyze the convergence process of steady flow field according to the solution vectors from the previous time steps. Unlike the traditional multigrid method, we project the flowfield solutions from the physical space into the modal space, and truncate all the high-frequency modes but only the first-order mode are retained based on the DMD analysis. The real solutions in the physical space can be obtained simply by the inverse transformation from the modal space. The developed MMG method ingeniously avoids the complicated process of coarsening computational mesh, and does not need to make any change for the grid in physical space. Therefore, it is very convenient to be applied to any numerical schemes with just little change for the flow solver, which is also suitable for unstructured grids and easy for parallel computing. Several typical test cases have been used to verify the effectiveness of the proposed method, which demonstrates that the MMG can dramatically reduce the number of iterative steps for the different mesh types, different accuracy of spatial discretization and different time-marching schemes. The method is 3 to 6 times faster than the original method while ensuring the computational accuracy.

Machine Learning in Aerodynamic Shape Optimization

“…Instead of the initial guess, Andersson [410] adopted DMD in intermediate iterations to correct the solution closer to a steady-state condition and thus accelerated the convergence of steady flow simulations. Liu et al [411] proposed a mode multigrid method using DMD to truncate all the high-frequency parts in steady flow simulations and achieved speedup ratios of 3 to 6 while ensuring the computational accuracy, and Chen et al [412] used a similar strategy to accelerate the convergence of the steady adjoint equations. Liu et al [413] improved the efficiency of implicit dual time scheme for unsteady flow simulations by providing proper initial solutions using DMD at each physical time step.…”

Unsteady 2D and 3D Navier-Stokes Solver with Application of Multigrid Scheme to Pressure Poisson Fractional Step on Arbitrary Unstructured Grids in Various Applications with Emphasis on Ship Motion

“…ey applied the presented method to the Hodge-Helmholtz decomposition test case and showed better accuracy related to second-order discretization. In 2019, Liu et al [23] proposed a mode multigrid method and applied it to a steady-state solver for unstructured grids. ey used the solver for 2D and 3D foils and showed that the method is 3 to 6 times faster than the baseline method while ensuring computational accuracy.…”