曲域上标量守恒律的合并单元间断伽辽金方法
A merged-cell discontinuous Galerkin method for scalar conservation laws on curved domains
浏览论文内容
中文总结 AI 辅助
本文提出并分析了一种用于曲域上标量守恒律的合并单元间断伽辽金方法，通过矩形合并处理切割单元，证明了半离散$L^2$耗散和最优误差估计，数值实验验证了接近$k+1$的收敛阶。
中文摘要 AI 辅助
我们分析了一种用于二维曲域上标量守恒律的间断伽辽金方法。笛卡尔网格与域的小交集通过矩形单元合并来处理。局部空间是$\nmathbb Q_k$在每个物理合并单元上的限制。熵守恒通量辅以面粘性，对于奇数次数，在未合并的内部正方形上还辅以斜导数跳跃修正。我们证明了精确的半离散$L^2$耗散性，以及对每个固定$k\ge1$的最优误差估计。对于在整个边界上具有齐次迹的光滑解，半离散方法满足$O(h^{k+1})$误差界。常数与原始切割单元面积分数无关。证明结合了物理单元$L^2$投影和规则面投影误差的消除。一个单独的半离散结果涵盖了具有迎风入流数据的线性输运。在圆盘上的Burgers方程和光滑星形上的线性平流计算显示，对于$k=1,2,3$，阶数接近$k+1$。
英文摘要
We analyse a discontinuous Galerkin method for two-dimensional scalar conservation laws on curved domains. Small intersections of a Cartesian grid with the domain are combined by rectangular cell merging. The local space is the restriction of $\mathbb Q_k$ to each physical merged cell. An entropy-conservative flux is supplemented by face viscosity and, for odd degrees, a skew derivative-jump correction on unmerged interior squares. We prove exact semidiscrete $L^2$ dissipation and an optimal error estimate for every fixed $k\ge1$. For smooth solutions with homogeneous trace on the entire boundary, the semi-discrete method satisfies an $O(h^{k+1})$ error bound. The constants are independent of the original cut-cell area fractions. The proof combines a physical-cell $L^2$ projection and cancellation of regular-face projection errors. A separate semidiscrete result covers linear transport with upwind inflow data. Computations for Burgers' equation on a disk and linear advection on a smooth star exhibit orders close to $k+1$ for $k=1,2,3$.
发表机构
- School of Mathematical Science, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
