Boundary corrections for kernel approximation to differential operators

TL;DR

This work develops high-order kernel-based differential operators on bounded domains by introducing generalized boundary terms within the $MOL^T$ framework. The authors construct boundary-corrected first and second derivative operators, derive their Neumann-series representations, and propose modified partial sums that incorporate boundary-derivative information to achieve $k$-th order accuracy for both $\partial_x$ and $\partial_{xx}$. They provide convergence analyses and implement practical numerical strategies (WENO/ENO quadrature and SSP-RK time integration) to realize the schemes on uniform and nonuniform grids, including Dirichlet and Neumann boundaries. Numerical experiments across 1D and 2D problems validate the theory, showing that the method attains the expected order on bounded domains and remains unconditionally stable, enabling large time steps in a variety of PDEs such as transport, wave, and convection–diffusion equations.

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.

Figures (7)

• Figure 4.1: Circular mesh along $x$-axis (blue) and $y$-axis (red).
• Figure 4.2: Nonuniform third order ENO method.
• Figure 5.3: The square nonuniform mesh used and numerical solution of Example \ref{['ex:squre_wave']} with third order scheme.
• Figure 5.4: Numerical solution of Example \ref{['ex:drumhead']} (2) with different time level with third order scheme.
• Figure 5.5: Numerical solution of Example \ref{['ex:drumhead']} (1) with the first eigen-mode (left) and Example \ref{['ex:drumhead']} (2) with the second eigen-mode (right) at $T = 1$.

Theorems & Definitions (19)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof