AI 中文总结
本文提出基于算子半群的半离散DG方法新框架，构造满足无穷多局部熵不等式的高阶OFDG型格式，推广至任意维度守恒律系统，证明了光滑解最优误差估计与严格凸守恒律间断解的强收敛性。
AI 中文摘要
针对标量守恒律，我们提出了一种基于算子半群推导半离散间断伽辽金（DG）方法的新框架，将其应用于构造一类高阶OFDG型格式[13]，这类格式在非均匀网格上结合通用E通量满足无穷多局部熵不等式。通过采用文献[1]中定义的熵稳定数值通量，该类格式可进一步推广到任意空间维度的守恒律系统。最后，我们证明了非线性标量守恒律光滑解的最优误差估计，并通过补偿紧致性方法证明了严格凸守恒律间断解的强收敛性。
英文摘要
We propose a novel framework for deriving semi-discrete discontinuous-Galerkin (DG) methods using operator semigroups for scalar conservation laws, then apply it to construct a class of high-order OFDG-type schemes [13] satisfying infinitely many local entropy inequalities with general E-fluxes on non-uniform meshes. Such schemes are further generalized to systems of conservation laws in any number of space dimensions by using entropy stable numerical fluxes in the sense of [1]. Finally, we prove optimal error estimates for smooth solutions to nonlinear scalar conservation laws, and prove strong convergence for discontinuous solutions to strictly convex conservation laws via compensated compactness.
