In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and optimal order of accuracy. Our lowest order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations, and show that our scheme is positivity preserving. In the presence of shocks additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

翻译：本文提出了一类具有保界性质的高阶切割有限元方法，用于求解一维空间双曲守恒律。该方法基于间断伽辽金框架，采用规则背景网格，允许内部边界任意切割网格单元。我们采用鬼罚稳定性处理小切割单元，并在宏单元（由切割单元与未切割单元通过稳定化连接的局部区域）上建立新的近似重构方法。研究表明，重构解保持守恒性和最优精度。最低阶格式得到的分片常数解满足标量双曲守恒律的最大值原理。将该最低阶格式应用于欧拉方程时，格式具有保正性，即保持压力和密度为正。对于高阶格式，我们在宏单元的重构解上施加合适的保界限制器。标量情形中采用最大值原理限制器，确保受限近似满足最大值原理；相应地，对欧拉方程采用保正限制器，并证明格式具有保正性。存在激波时需额外限制以避免振荡，因此我们对重构解应用标准TVB限制器。时间步长限制与背景网格上对应间断伽辽金方法的阶数相同。数值算例验证了所提格式的精度、保界性和激波捕捉能力。