高斯 FSBP 算子：双曲守恒律数值方法的比较与应用
Gaussian FSBP operators: Comparison and application to numerical methods for hyperbolic conservation laws
AI总结：
本文比较用广义高斯求积构造的开放和封闭 FSBP 算子，将其应用于双曲守恒律数值求解，并扩展 FSBP 框架引入外推算子。通过一维线性平流等方程的数值实验，验证该方法能提高效率和精度，在时间相关设置中有优势。
AI中文摘要：
函数空间分部求和（FSBP）算子能实现基于一般非多项式逼近空间的双曲守恒律的守恒和能量稳定数值方法。近期研究表明，与大多聚焦等距网格的现有构造相比，使用广义高斯求积显著减少所需网格点数。本文比较用广义高斯求积构造的开放和封闭 FSBP 算子，并将其应用于数值求解双曲守恒律。为支持开放节点分布，通过引入函数空间精确外推算子扩展 FSBP 框架，并在求解双曲守恒律的数值格式中实现。数值实验包括一维线性平流、无粘伯格斯方程和气体动力学的可压缩欧拉方程。观察到在数值格式中应用 FSBP 算子可提高效率和精度，且在更具挑战性的时间相关设置中展现出优势。
英文摘要：
Function-space summation-by-parts (FSBP) operators enable conservative and energy-stable numerical methods for hyperbolic conservation laws based on general, non-polynomial approximation spaces. Recent works show that using generalized Gaussian quadrature significantly reduces the number of grid points required compared to existing constructions that have mostly focused on equidistant grids. In this paper, we compare open and closed FSBP operators constructed with generalized Gaussian quadratures and apply them to numerically solve hyperbolic conservation laws. Furthermore, to support open node distributions, we extend the FSBP framework by introducing function-space exact extrapolation operators and operationalize them in numerical schemes for solving hyperbolic conservation laws. Our numerical experiments include the one-dimensional linear advection, non-viscous Burgers, and compressible Euler equations of gas dynamics. We observe that applying FSBP operators in numerical schemes can improve efficiency and accuracy. Notably, we demonstrate these advantages in more challenging time-dependent settings compared to other recent works on Gaussian FSBP operators.
