A finite volume method with Walsh basis functions to capture discontinuity inside grid

The traditional finite volume or finite element method assumes that the flow variables are continuous in the control volume, and the position of discontinuity is restricted to the interface of the control volume, therefore it is impossible to capture discontinuity inside a control volume. In this paper, the hypothesis that the flow variables are continuous in the control volume is abandoned. The Walsh basis functions constituted by square waves are applied to represent the conservative variables in a control volume with discontinuous forms rather than the traditional continuous forms. According to the positions of discontinuities contained in the Walsh approximation forms of conservative variables which are introduced by the Walsh functions, the control volume can be divided into series of virtual sub-cells. Integrating and solving the conservative equations represented by Walsh basis function coefficients on each sub-cell, the discontinuity can be captured inside a control volume. This solving method is named as "Finite volume method with Walsh basis functions". Compared with the traditional finite volume method, this method can reduce the numerical errors by a certain proportion and improve the resolution of capturing discontinuities. While for sub-cell scale, this method has only first-order calculation accuracy. In order to further improve the resolution of the smooth solutions, the linear / nonlinear approximations can be reconstructed by using the sub-cell average values of conservative variables in each control volume to realize second order / higher order calculation accuracy. Finally, in numerical tests, the finite volume method based on Walsh basis functions is used to solve several typical unsteady problems of inviscid Burgers equation and Euler equations with respect to one-dimensional and two-dimensional cases. By comparing the obtained numerical results of the new method and the traditional finite volume method, the accuracy, efficiency, robustness and the ability of capturing discontinuity of the proposed method are verified. © 2021, Chinese Journal of Theoretical and Applied Mechanics Press. All right reserved.