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 引用本文: 王刚, 干源, 任炯. Walsh函数有限体积法的多重网格特征研究. 力学学报, 2022, 54(12): 3418-3429.
Wang Gang, Gan Yuan, Ren Jiong. Investigation on multigrid features of the finite volume method with Walsh basis function. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3418-3429.
 Citation: Wang Gang, Gan Yuan, Ren Jiong. Investigation on multigrid features of the finite volume method with Walsh basis function. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3418-3429.

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

• 摘要: Walsh函数有限体积法(FVM-WBF)是一种能够在网格内部捕捉间断的新型数值方法. 持续增加Walsh基函数数目能够稳步提高FVM-WBF方法的求解分辨率, 但计算量暴发式增长和收敛速度下降的问题也会同步出现. 针对Walsh基函数数目增加而引起的计算效率问题, 本文分析了Walsh基函数及其系数所能影响的网格单元局部均值区域尺度, 发现其中隐含类似多重网格的尺度特征, 据此提出一种结合多重网格策略的FVM-WBF方法. 在定常流场计算中根据各级Walsh基函数影响尺度的不同, 对每级Walsh基函数设置满足其稳定性约束的时间步长, 在时间推进求解的过程中快速消除不同波长的数值误差, 实现多重网格的加速收敛效果. 选取NACA0012翼型和二维圆柱的定常无黏绕流问题作为算例, 对引入多重网格策略的FVM-WBF方法和不考虑多重网格策略的FVM-WBF方法进行对比测试. 数值结果证实: 新发展的FVM-WBF方法具备多重网格的关键特征, 在不增加任何特殊处理和计算量的情况下, 只需通过时间步长的调整, 就能够达到多重网格的加速效果, 显著提升计算效率.

Abstract: 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.

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