A novel central compact finite-difference scheme for third derivatives with high spectral resolution

TL;DR

This paper presents a new high-order central compact finite-difference scheme for third derivatives that achieves superior spectral resolution and low dissipation, improving the numerical solution of dispersive wave equations.

Contribution

The paper introduces a novel compact scheme combining cell-node and cell-centered approaches to accurately compute third derivatives with high spectral resolution.

Findings

Exhibits high order accuracy and superior spectral resolution.

Reduces transfer errors compared to conventional schemes.

Effective in solving dispersive wave equations with third derivatives.

Abstract

In this paper, we introduce a novel category of central compact schemes inspired by existing cell-node and cell-centered compact finite difference schemes, that offer a superior spectral resolution for solving the dispersive wave equation. In our approach, we leverage both the function values at the cell nodes and cell centers to calculate third-order spatial derivatives at the cell nodes. To compute spatial derivatives at the cell centers, we employ a technique that involves half-shifting the indices within the formula initially designed for the cell-nodes. In contrast to the conventional compact interpolation scheme, our proposed method effectively sidesteps the introduction of transfer errors. We employ the Taylor-series expansion-based method to calculate the finite difference coefficients. By conducting systematic Fourier analysis and numerical tests, we note that the methods exhibit exceptional characteristics such as high order, superior resolution, and low dissipation. Computational findings further illustrate the effectiveness of high-order compact schemes, particularly in addressing problems with a third derivative term.