Kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same computational cost as an explicit finite difference scheme but can exhibit order reduction at boundaries. In previous work on periodic domains, [8,9], order reduction was addressed, yielding high-order accuracy. The issue addressed in this work is the elimination of order reduction of the kernel-based approach for a more general set of boundary conditions. Further, we consider the case of both first and second order operators. To demonstrate the theory, we provide not only the mathematical proofs but also experimental results by applying various boundary conditions to different types of equations. The results agree with the theory, demonstrating a systematic path to high order for kernel-based methods on bounded domains.

翻译：基于核的偏微分方程算子逼近方法已被证明对线性偏微分方程具有无条件稳定性，并在数值实验中表现出对非线性偏微分方程的无条件稳定性。这些方法的计算成本与显式有限差分格式相当，但在边界处可能出现阶数退化。在先前关于周期域的研究[8,9]中，通过修正解决了阶数退化问题，实现了高阶精度。本文旨在针对更一般的边界条件集合，消除基于核的方法的阶数退化现象。此外，我们同时考虑一阶和二阶算子的情形。为验证理论，我们不仅提供了数学证明，还通过对不同类型方程施加多种边界条件进行了实验验证。实验结果与理论预测一致，为有界域上基于核的方法实现高阶精度提供了系统化路径。