In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the $L^\infty$-in-time $L^2$-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results.

翻译：本文针对二维直坡道上的切割单元网格求解非定常对流方程，提出了一种先验误差分析。空间离散采用最低阶迎风型间断伽辽金格式，其中引入\textit{依赖域}（DoD）稳定化技术来修正小切割单元邻域内的更新。由此，可采用显式时间推进格式，其时间步长与极小切割单元的尺寸无关。我们的误差分析基于一阶线性偏微分方程误差估计的一般框架，该框架依赖于离散双线性形式的一致性、有界性和离散耗散性。我们证明了包含DoD稳定化的空间离散格式满足这些性质。这使得我们能够对全离散格式，在结合了时间$L^\infty$-空间$L^2$范数与包含速度加权跳跃的半范数的度量下，证明具有二分之一阶的拟最优误差估计。文中也给出了相应的数值结果。