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. 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含时偏微分方程。求和分部算子对于开发此类问题的高阶精度稳定数值方法至关重要。传统上，SBP算子基于多项式能精确逼近解的假设进行定制，因此SBP算子需对该假设成立。然而，这一假设对于诸多更适合其他逼近空间的问题而言存在不足。我们近期解决了这一问题，并发展了基于通用函数空间的一阶导数SBP算子理论，将其命名为函数空间SBP算子。本文进一步将FSBP算子的创新拓展至二阶导数。所开发的二阶导数FSBP算子既保留了现有多项式SBP算子所需的模仿特性，又通过适用于更广泛函数空间而实现了更高灵活性。我们证明了此类算子的存在性，并详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多种函数空间，我们展示了该方法的普适性。本研究为基于适当函数空间的二阶导数SBP算子的应用开辟了可能，为未来广泛应用铺平了道路。