自适应Walsh函数有限体积方法

Adaptive finite volume method with Walsh basis functions

Walsh函数有限体积方法(FVM-WBF)是一种具备网格内部捕捉间断能力的新型数值方法。全局增加网格单元内Walsh基函数的数目可以有效提升FVM-WBF方法的数值分辨率,但同时会带来计算量的急剧增加。为了平衡FVM-WBF方法分辨率和计算效率之间的矛盾,利用Walsh基函数的数值特性,提出了一种结合自适应策略的Walsh函数有限体积方法。该方法根据流场特征动态调整网格单元内的Walsh基函数数目,仅在流场结构变化剧烈的局部区域使用足量的基函数,避免全局性增加基函数所引发的计算量爆发式增长。选取二维非定常双马赫反射、RayleighTaylor不稳定性问题和定常NACA0012翼型绕流为算例,将新发展的自适应Walsh函数有限体积方法与原始Walsh函数有限体积方法进行对比测试。理论分析和数值结果表明,Walsh函数有限体积方法“天然”具备便捷的自适应潜能,能够方便地依据流场特征动态地实现Walsh基函数数目的优化配置,实现高分辨率和高效率的双赢。

The Finite Volume Method with Walsh Basis Functions(FVM-WBF method)is a novel numerical method with the ability to capture discontinuity inside grids.While globally increasing the number of Walsh basis functions can effectively improve the numerical resolution,it also leads to a large increase in computational costs.To balance the resolution and the computational efficiency of the FVM-WBF method,an adaptive finite volume method with Walsh ba⁃sis functions is proposed according to the numerical properties of the Walsh basis functions.The proposed method dy⁃namically adjusts the number of Walsh basis functions in the grid based on the features of the flow field.Sufficient ba⁃sis functions are employed only in the local region where the flow field structure changes dramatically,so as to avoid the explosive growth of the computation caused by the global increase of the basis functions.Several cases are se⁃lected to test the adaptive FVM-WBF method in comparison with the original FVM-WBF method,including twodimensional Double Mach reflection problem,Rayleigh-Taylor instability problem and the flow over NACA0012 airfoil.Theoretical analysis and numerical results show that the adaptive FVM-WBF method has the inherent capability of con⁃venient dynamic adaptation,and a balance between high resolution and high efficiency in the numerical simulations has been achieved by intelligently adapting the number of Walsh basis functions in the flow field.