Level set (LS) method is a widely used interface capturing method. In the simulations of incompressible two-phase flows, in order to avoid discontinuities at interfaces, the LS function is usually taken as a smeared-out Heaviside function bounded on [0, 1] and advected by a given velocity field \(\mathbf {u}\) obtained from the solution of the incompressible Navier-Stokes equations. In the incompressible limit \(\nabla \cdot \mathbf {u}=0\), the advection equation for the LS function can be written and discretized in conservative form. However, due to numerical errors, the resulting velocity field is in general not divergence free which leads to the solution of the advection equation in conservative form does not satisfy the maximum principle. To overcome this issue, in this work, we develop a high-order discontinuous Galerkin (DG) method to directly solve the advection equation for the LS function in non-conservative form. Moreover, we prove that by applying a linear scaling limiter, the proposed method together with a strong stability preserving (SSP) time discretization scheme can satisfy the strict maximum principle under a suitable CFL condition. Numerical simulations of several well-known benchmark problems, including the application to incompressible two-phase flows, are presented to demonstrate the high-order accuracy and maximum-principle-satisfying property of the proposed method.

水平集（LS）方法是一种广泛使用的接口捕获方法。在不可压缩两相流的模拟中，为了避免界面不连续，通常将LS函数视为以[0，1]为边界并由给定速度场\（\ mathbf { u} \）是从不可压缩的Navier-Stokes方程的解中获得的。在不可压缩的极限\（\ nabla \ cdot \ mathbf {u} = 0 \），可以用保守的形式写和离散LS函数的对流方程。然而，由于数值误差，所得到的速度场通常不是无散度的，这导致对流方程的保守形式的解不能满足最大原理。为了克服这个问题，在这项工作中，我们开发了一种高阶不连续伽勒金（DG）方法，以非保守形式直接求解LS函数的对流方程。此外，我们证明了通过应用线性比例限制器，该方法与强稳定度（SSP）时间离散方案一起可以在适当的CFL条件下满足严格的最大原理。几个著名基准问题的数值模拟，包括对不可压缩的两相流的应用，