A novel central compact finite-difference scheme for third derivatives with high spectral resolution
Lavanya V Salian, Samala Rathan, Debojyoti Ghosh
TL;DR
This work addresses the challenge of accurately resolving the third-derivative term in dispersive PDEs, such as the KdV equation, by introducing a new central compact scheme (TDCCS) that combines cell-node and cell-center data. Derivations of coefficients via Taylor expansion (TE) and least-squares optimization (LS) yield high-order families (E/T/P) and a CI-enhanced variant, with Fourier analysis confirming superior spectral resolution and minimal dispersion. Numerical experiments demonstrate that TDCCS achieves higher accuracy and spectral-like resolution than traditional cell-node schemes, including in nonlinear soliton and dispersive limits, though at increased memory usage and tighter time-step constraints. The method offers a powerful tool for direct numerical simulations of dispersive waves, with potential improvements via optimized time integrators and filtering strategies to balance accuracy, stability, and efficiency.
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.