Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. We showcase the superior performance of these non-polynomial FSBP operators over traditional polynomial-based operators for a suite of one- and two-dimensional problems, encompassing a boundary layer problem and the viscous Burgers' equation. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.

翻译：许多应用依赖于求解包含二阶导数的时间相关偏微分方程（PDE）。求和-积（SBP）算子是开发此类问题稳定、高精度数值方法的关键。传统上，SBP算子基于多项式能精确逼近解的假设而定制，因此应针对多项式具有精确性。然而，这一假设对于一系列其他近似空间更适用的问题并不充分。我们近期解决了这一问题，并发展了基于一般函数空间的一阶导数SBP算子理论，称为函数空间SBP（FSBP）算子。在本文中，我们将FSBP算子的创新扩展到二阶导数。所发展的二阶导数FSBP算子保留了现有多项式SBP算子的期望仿射性质，同时通过适用于更广泛的函数空间而实现更大灵活性。我们证明了这些算子的存在性，并详细阐述了构建它们的直接方法。通过探索包括三角函数、指数函数和径向基函数在内的各种函数空间，我们展示了该方法的通用性。我们通过一系列一维和二维问题（包括边界层问题和粘性Burgers方程）的数值实验，证明了这些非多项式FSBP算子相较于传统多项式算子的优越性能。本文的研究为基于合适函数空间使用二阶导数SBP算子开辟了可能性，为未来广泛应用铺平道路。