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 order of convergence. 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：本文提出了一族具有保界性质的高阶切割有限元方法，用于求解一维空间中的双曲守恒律。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。我们的方法包含用于处理小切割单元的幽灵罚稳定化技术，以及在宏单元上对近似解进行重构的新策略；宏单元是由通过稳定化相互连接的切割与非切割相邻单元组成的局部单元块。我们证明了重构解保持守恒性并具有收敛阶。我们的最低阶格式产生分段常数解，该解满足标量双曲守恒律的最大值原理。当最低阶格式应用于欧拉方程时，该格式具有保正性，即压力和密度的正值性得以保持。对于高阶格式，我们在宏单元上对重构解施加适当的保界限制器。在标量情形中，采用最大值原理限制器，确保受限近似解满足最大值原理。相应地，对于欧拉方程我们采用保正限制器，并证明该格式具有保正性。在存在激波的情况下，需要额外的限制以避免振荡，因此我们对重构解应用标准的TVB限制器。时间步长限制与背景网格上相应间断伽辽金方法的限制同阶。数值计算验证了所提格式的精度、保界特性以及激波捕捉能力。