AI 中文总结
本文扩展了现有SBP离散格式的收敛分析，将适用范围扩展至含时空依赖源项的一般双曲系统，证明其对光滑解的收敛性，数值结果验证了收敛速率的尖锐性。
AI 中文摘要
尽管双曲守恒律的基于熵的求和分部（Summation-by-Parts，SBP）离散格式因鲁棒性和稳定性而被广泛应用，但关于其收敛性的结果却很少。我们从两个方面扩展了Worku、Del Rey Fernández和Zingg（2026，DOI：https://doi.org/10.48550/arXiv.2603.18369）的最新收敛分析：第一，不再仅考虑通量为齐次且二阶导数全局有界的双曲守恒律（该限制本质上仅适用于线性或二次通量），而是考虑具有严格凸熵且源项依赖时间和空间的一般双曲系统；第二，不再要求特殊类别的SBP算子，而是考虑曲面上的对角范数SBP算子的一般框架，包括有限差分法、连续和间断Galerkin法。由于误差分析基于离散相对熵，因此仅适用于光滑解。为实现守恒律的统一处理，我们将分析限制为周期边界条件。数值结果表明，预测的收敛速率总体上是尖锐的，但对于特殊情况可改进，如偶多项式阶的间断Galerkin法和多块有限差分法。最优分析预计需要更复杂的、针对特定方法类别的论证，而非本工作中使用的SBP算子一般框架。
英文摘要
Although entropy-based summation-by-parts (SBP) discretizations of hyperbolic conservation laws are widely used for their robustness and stability properties, there are very few results on their convergence. We extend a recent convergence analysis of Worku, Del Rey Fernández, and Zingg (2026, DOI: 10.48550/arXiv.2603.18369) in two ways. First, instead of allowing only hyperbolic conservation laws whose fluxes are homogeneous and have globally bounded second derivatives (a restriction essentially to linear or quadratic fluxes), we consider general hyperbolic systems with strictly convex entropy and source terms depending on time and space. Second, instead of requiring a special class of SBP operators, we consider a general framework of diagonal-norm SBP operators on curved meshes, including finite differences, continuous and discontinuous Galerkin methods. Since the error analysis is based on a discrete relative entropy, it is restricted to smooth solutions. To enable a unified treatment of conservation laws, we restrict the analysis to periodic boundary conditions. Numerical results demonstrate that the predicted convergence rates are sharp in general, but can be improved for special cases such as discontinuous Galerkin methods with even polynomial degree and multi-block finite difference methods. An optimal analysis is expected to require more sophisticated arguments specialized to the class of methods instead of the general framework of SBP operators used in this work.
CommentsAll code required to reproduce the numerical results is available online at https://github.com/ranocha/2026_convergence_sbp and https://zenodo.org/doi/10.5281/zenodo.21672279
