## 1 Introduction
However, great efforts have to be input to design such techniques. In order to achieve high accuracy near discontinuities, many non-oscillation schemes have been proposed. For example, Monotone Upstream-centered Schemes for Conservation Law (MUSCL), a kind of Total Variation Diminishing (TVD) schemes, avoids numerical oscillations by adding a flow limiter [9]. TVD schemes ensures that the physical field in the transition region is monotonic and bounded, but it does not solve the problem of numerical dissipation. Numerical manifold methods (NMM) have been invented and applied to numerical computation and engineering by [PERSON] et al. [10]. NMM are capable of dealing with discontinuities in a unified way. In NMM, the mathematical cover can be cut into many physical patches by the solution domain, while the local approximation is constructed directly on the physical patches. The function that constructs the local approximation can be discontinuous, so the numerical manifold method can handle discontinuity problems in a natural way. So far, many problems have been well solved by NMM, such as the seepage problems [11], the wave propagation problems [12], the two-dimensional dynamic crack propagation problems [13; 14; 15], the three-dimensional crack propagation problems [16], the fourth-order problem [17], the contact problem [18], and so on. It is well known that the analytic solution can be viewed as a higher-order polynomial. The higher the order of expansion, the higher the accuracy of the approximation. The NMM is also based on the partition of unit scheme, the global approximation can be established by multiplying a weight function by local approximations. Like generalized finite elements, numerical manifold methods can construct higher order global approximation by simply raising the order of the local approximations or weight function. Currently, there are two main skills for constructing high accuracy global approximations [19]. At present, the numerical manifold element method is already a very effective method in the solid field, and it is increasingly used in the field of engineering simulation. For example, hydraulic fracturing [20; 21], pore medium problem [22; 23; 24], dynamic consolidation [25], fracture problem [26], slope stability analysis [27]. Although there are a few studies on the numerical manifold element method in fluid mechanics, it has been successfully applied to solve the incompressible and viscous Navier-Stokes equation [28; 29]. However, there are no articles about applying numerical manifold element method to fluid discontinuities. Numerical manifold element method has a universal way in solving discontinuous problems, so it is very worthwhile to develop it in the field of fluid mechanics. Facing fluid mechanics problems with the discontinuity, the interpolation of FDM or FVM will result in the slope at discontinuity. However, the NMM can show great potential to approximate the discontinuity, making it is necessary to develop NMM in fluid mechanics. This paper is an attempt of the numerical manifold element method in the field of computational fluid dynamics with discontinuities. The main contents of this manuscript are as follows. In Section 2, the foundation of the numerical manifold method is briefly introduced. In Section 3, the construction of local and global approximations is expounded upon. The new scheme for singular patch reconstruction is presented in detail. In Section 4, the discretization of equations is carried out. In Section 5, numerical results of some tests are given and compared with other schemes. The linear dependent issue will be discussed in detail. Some remarks, conclusions, and prospects are presented in Section 6.