Abstract:

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 basis functions is proposed according to the numerical properties of the Walsh basis functions. The proposed method dynamically adjusts the number of Walsh basis functions in the grid based on the features of the flow field. Sufficient basis 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 selected to test the adaptive FVM-WBF method in comparison with the original FVM-WBF method， including two-dimensional 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 convenient 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.

Key words: computational fluid dynamics, finite volume method, Walsh functions, adaptive method, discontinuous capture