Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.
翻译:逐个总结(SBP)操作员使我们能够系统地为基于时间的差别方程式制定能源稳定和高顺序精确的精确数字方法。直到最近,现有的SBP操作员的主要想法是,多球类操作员能够准确地接近解决方案,因此,SBP操作员应该对此十分精确。然而,多球类操作员不能为一些问题提供最佳近似近似,而其他近似空间则更为合适。我们最近讨论了这一问题,并为一维的SBP操作员开发了一个理论,其基础是一般功能空间、硬体功能-空间SBP(FSBP)操作员。在本文中,我们将FSBP操作员的理论扩大到多个层面。我们注重于其存在、与二次、建筑和模拟特性的连接。多球类运行员及其应用的更详尽数字演示将在未来工作中提供。类似一维案例,我们证明,基于多球类的SBPP(MS BBP)操作员的既定结果大部分可以延续到较普通的MFSBPBP操作员类别。我们的结论结论显示,SBPA级操作员的数值方法可以比已经增加。SBPBPBPA级操作员的数值方法。