高阶流线扩散的正规方程与离散能量结构
Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion
浏览论文内容
中文总结 AI 辅助
本文首次对高阶线性多步格式与流线扩散有限元结合的多维一阶双曲系统进行稳定性与误差分析，通过正规方程结构实现最优收敛阶。
中文摘要 AI 辅助
我们分析了用于具有斜对称空间算子的时间依赖一阶系统的全离散流线扩散有限元方法，也称为SUPG方法。据我们所知，这是首次对强一致的流线扩散有限元与阶数大于2的线性多步格式相结合的多维一阶双曲系统进行稳定性和先验误差分析。局部时间$L^2$残差最小化确定了$\u03b4=b_0\u03c4$，其中$\u03b4$是稳定化参数，$\u03c4$是时间步长，$b_0$是方法平均值中的当前时间系数。对于这一选择，每个隐式线性系统都是图范数中的对称正定正规方程。直接的对偶残差估计得出误差阶为$O(h^{k+1/2}+\u03c4^\u03bd)$，当$\u03b4$与$h$成正比且$\u03c4=O(h)$时，其中$h$是网格尺寸，$k$是多项式次数，$\u03bd$是时间阶。对$\u03b8$-方法和Adams-Moulton方法AM3-AM5建立了稳定性。一般次数的AM3和AM4估计使用加强的CFL条件；对于连续分段仿射单元，单元级相消恢复了标准双曲区域。AM5和显式Adams-Bashforth方法AB3和AB4在标准双曲CFL条件下是稳定的，尽管显式格式不具有正规方程结构。阶匹配的声学测试恢复了二阶到五阶以及低半阶的材料残差率。在具有不连续初始数据的d'Alembert测试中，正规方程SUPG在波前之外恢复了二阶局部$\u0050_1$收敛，而未稳定化的误差接近半阶。二维压缩波测试显示了改进的局部化，具有适度的耗散。
英文摘要
We analyse fully discrete Streamline Diffusion finite element methods, also known as SUPG methods, for time-dependent first-order systems with skew-symmetric spatial operators. To the best of our knowledge, this is the first stability and a priori error analysis of strongly consistent Streamline Diffusion finite elements combined with linear multistep schemes of order greater than two for multidimensional first-order hyperbolic systems. Local-in-time $L^2$-residual minimisation singles out $δ=b_0τ$, where $δ$ is the stabilisation parameter, $τ$ the time step, and $b_0$ the current-time coefficient in the method average. For this choice, each implicit linear system is a symmetric positive definite normal equation in a graph norm. A direct dual-residual estimate yields errors of order $O(h^{k+1/2}+τ^ν)$ when $δ$ is proportional to $h$ and $τ=O(h)$, where $h$ is the mesh size, $k$ the polynomial degree, and $ν$ the temporal order. Stability is established for the $θ$-method and the Adams-Moulton methods AM3-AM5. The general-degree AM3 and AM4 estimates use strengthened CFL conditions; for continuous piecewise affine elements, an elementwise cancellation recovers the standard hyperbolic regime. AM5 and the explicit Adams-Bashforth methods AB3 and AB4 are stable under a standard hyperbolic CFL condition, though the explicit schemes do not have normal-equation structure. Order-matched acoustic tests recover orders two through five and material-residual rates one half-order lower. In a d'Alembert test with discontinuous initial data, normal-equation SUPG recovers second-order local $\mathbb P_1$ convergence away from the wave fronts, whereas the unstabilised error is close to half order. A two-dimensional compact-wave test shows improved localisation with modest dissipation.
