INVESTIGATION ON MULTIGRID FEATURES OF THE FINITE VOLUME METHOD WITH WALSH BASIS FUNCTION

The finite volume method with Walsh basis functions (FVM-WBF) is a novel numerical method with the ability to capture discontinuity inside grid. The numerical resolution of the FVM-WBF method can be effectively improved by increasing the number of basis functions, but the explosive growth of computation and the decrease of convergence speed will also appear simultaneously. To relieve the computational costs due to the increasing of the number of basis functions, the scales of the piecewise continuous mean value subdomains inside the grid cell, which are dominated by different levels of Walsh basis functions and their coefficients, have been analyzed. It is found that FVM-WBF method implicitly has scale characteristics similar to multigrid. Based on this finding, an FVM-WBF method combined with multigrid strategy is presented. In time integration stage of steady flow simulation, this newly developed FVM-WBF method defines the maximal time step for each level of Walsh basis function according to their influence scales and the corresponding numerical scheme stability constraint. As a result, the numerical error of different wavelengths in the process of time advancing is quickly eliminated and the convergence can be accelerated. Several test cases are selected to evaluate the multigrid features of the presented FVM-WBF method, including the low speed inviscid flow over two-dimensional cylinder and a set of inviscid steady flow with different Mach number around NACA0012 airfoil. The numerical results confirm that the newly developed FVM-WBF method has the key characteristics of multigrid, and convergence rate can be greatly enhanced only by adjusting the time step without any additional processing and computational costs.