标题： 最优精确算子用于偏微分方程 显示英文标题

标题： Optimally accurate operators for partial differential equations

摘要： 在本文中，我们推广了在一系列关于地震波传播模拟的研究中提出并使用的\textit{最优精确算子}概念，特别是在 Geller \& Takeuchi (1995) 的研究基础上。 尽管这些算子在数学和数值上已经被证明比传统方法更准确，但该理论是专门针对线性弹性连续介质的动力学方程开发的。 此外，原始理论要求由于低阶截断导致的每一项误差补偿，在误差估计过程中这限制了将其应用到由偏微分方程描述的其他类型的物理现象。 在这里，我们提出了一种新的方法，可以自动推导任意偏微分方程的数值算子。 这些涉及少量时间和空间节点（紧凑型算子）的算子比传统的更准确，并且不需要网格划分。 我们的方法评估了以泰勒展开为辅助工具发展起来的动力学方程的弱形式。 我们建立了新方法与经典最优精度算子之间的联系，表明它们在均匀介质中产生相同的系数。 最后，我们在各种非均匀介质中进行了1D泊松问题的基准测试。 基准测试展示了我们的方法相比传统算子的优越性，即使使用一组线性B样条测试函数（三点帽函数）时也是如此。 然而，收敛率可能依赖于材料属性的波长：当材料属性具有与场相同的波长时，收敛率为O(4)，而对于其他模型，它可能较不高效，为O(2)。 显示英文摘要

摘要： In this contribution, we generalize the concept of \textit{optimally accurate operators} proposed and used in a series of studies on the simulation of seismic wave propagation, particularly based on Geller \& Takeuchi (1995). Although these operators have been mathematically and numerically proven to be more accurate than conventional methods, the theory was specifically developed for the equations of motion in linear elastic continuous media. Furthermore, the original theory requires compensation for errors from each term due to truncation at low orders during the error estimation, which has limited its application to other types of physics described by partial differential equations. Here, we present a new method that can automatically derive numerical operators for arbitrary partial differential equations. These operators, which involve a small number of nodes in time and space (compact operators), are more accurate than conventional ones and do not require meshing. Our method evaluates the weak formulation of the equations of motion, developed with the aid of Taylor expansions. We establish the link between our new method and the classic optimally accurate operators, showing that they produce identical coefficients in homogeneous media. Finally, we perform a benchmark test for the 1D Poisson problem across various heterogeneous media. The benchmarks demonstrate the superiority of our method compared to conventional operators, even when using a set of linear B-spline test functions (three-point hat functions). However, the convergence rate can depend on the wavelength of the material property: when the material property has the same wavelength as that of the field, the convergence rate is O(4), whereas it can be less efficient O(2) for other models.