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 引用本文: 任炯, 王刚. 一种在网格内部捕捉间断的Walsh函数有限体积方法[J]. 力学学报, 2021, 53(3): 773-788.
Ren Jiong, Wang Gang. A FINITE VOLUME METHOD WITH WALSH BASIS FUNCTIONS TO CAPTURE DISCONTINUITY INSIDE GRID[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 773-788.
 Citation: Ren Jiong, Wang Gang. A FINITE VOLUME METHOD WITH WALSH BASIS FUNCTIONS TO CAPTURE DISCONTINUITY INSIDE GRID[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 773-788.

## A FINITE VOLUME METHOD WITH WALSH BASIS FUNCTIONS TO CAPTURE DISCONTINUITY INSIDE GRID

• 摘要: 传统有限体积或有限元方法假定流动变量在单元内连续, 间断仅限于控制体的交界面上, 因此它们无法在控制体内部捕捉间断. 本文摒弃控制体内流动变量连续的假设, 将自身具有间断特点的Walsh基函数应用于有限体积方法, 把控制体内的流场变量表示成间断基函数的组合形式. 按照Walsh基函数在控制体内引入的间断数目和位置, 将控制体单元虚分为若干个分片连续的子单元, 并将Walsh基函数级数表征的守恒型控制方程在每个子单元上进行数值积分和离散求解.相对于传统有限体积方法, 这种利用Walsh基函数构造的新型有限体积方法能够以一定的比例减小数值误差, 提高分辨率, 并可实现控制体单元内部的间断捕捉, 本文将其命名为Walsh函数有限体积方法. 该方法在子单元尺度上仅具有一阶计算精度, 为进一步提高对光滑解的分辨率, 在每个控制体内利用子单元上的变量平均值进行重构, 提出了子单元尺度上具有的二阶/高阶计算精度的Walsh函数有限体积方法. 最后, 运用新发展的方法求解无黏Burgers方程和Euler方程, 并在相同的计算网格上与传统有限体积方法进行对比计算, 对新方法的计算精度、计算效率、间断捕捉能力和鲁棒性进行了验证.

Abstract: 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.

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