Boundary corrections for kernel approximation to differential operators

TL;DR

This paper develops boundary correction techniques for kernel-based differential operator approximations, enabling high-order accuracy on bounded domains with various boundary conditions, extending previous work on periodic domains.

Contribution

It introduces a systematic method to eliminate order reduction in kernel-based PDE approximations for general boundary conditions, including first and second order operators.

Findings

Theoretical proofs confirm high-order accuracy with boundary corrections.

Experimental results validate the effectiveness of the boundary correction methods.

Methods are applicable to various boundary conditions and PDE types.

Abstract

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.