发表机构
University of Delaware（特拉华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对高维输运方程的分片常数稀疏网格DG方法，证明了其尖锐CFL条件，并发现相比全网格方法可扩大时间步长，且给出了显式判据与数值验证。
AI 中文摘要
我们建立了分片常数稀疏网格间断伽辽金（DG）方法在向前欧拉时间步进下的尖锐CFL条件，该方法应用于任意维周期域上具有常系数的输运方程。对于输运速度$\boldsymbol c=(c_1,\ldots,c_d)$和大小为$h$的均匀网格，我们证明该格式是$L^2$稳定的当且仅当$\Delta t \leq {h}/{\max_{1\leq \ell\leq d}|c_\ell|}$，而相应的全网格迎风格式众所周知需要$\Delta t\leq h/\sum_{\ell=1}^d |c_\ell|$。因此，稀疏网格离散将可允许的时间步长扩大了因子${(\sum_{\ell=1}^d |c_\ell|)}/{(\max_{1\leq \ell\leq d}|c_\ell|)}$，该因子介于1和$d$之间，对于各向同性输运达到$d$。充分性的证明依赖于多水平Haar分解的投影泄漏恒等式，这些恒等式允许能量估计中的混合方向项被稀疏网格投影所丢弃的能量吸收。尖锐性的证明则基于最细水平下一维的交替模式。作为副产品，我们获得了$L^2$算子范数和放大算子谱半径的显式公式。对于一般向下封闭指标集上的空间，我们推导了显式的充分CFL条件及其尖锐性的几何判据。在二维和四维中的数值实验证实了理论结果。
英文摘要
We establish the sharp CFL condition for the piecewise constant sparse grid discontinuous Galerkin (DG) method with forward Euler time stepping, applied to transport equations with constant coefficients on periodic domains in arbitrary dimensions. For the transport velocity $\boldsymbol c=(c_1,\ldots,c_d)$ and a uniform mesh of size $h$, we prove that the scheme is $L^2$ stable if and only if $Δt \leq {h}/{\max_{1\leq \ell\leq d}|c_\ell|}$, whereas the corresponding full grid upwind scheme is well-known to require $Δt\leq h/\sum_{\ell=1}^d |c_\ell|$. The sparse grid discretization therefore enlarges the admissible time step by a factor of ${(\sum_{\ell=1}^d |c_\ell|)}/{(\max_{1\leq \ell\leq d}|c_\ell|)}$, which lies between $1$ and $d$ and reaches $d$ for isotropic transport. The proof of sufficiency relies on projection leakage identities for the multilevel Haar decomposition, which allow the mixed directional terms in the energy estimate to be absorbed by the energy discarded by the sparse grid projection. The proof of sharpness follows from alternating modes in one dimension at the finest level. As a by-product, we obtain explicit formulas for the $L^2$ operator norm and the spectral radius of the amplification operator. For spaces over general downward closed index sets, we derive an explicit sufficient CFL condition and a geometric criterion for its sharpness. Numerical experiments in two and four dimensions confirm the theoretical results.
