一种具有高谱分辨率的求解三阶导数的中心紧致差分新格式
A novel central compact finite-difference scheme for third derivatives with high spectral resolution
AI总结：
本文提出一类新的中心紧致差分格式，结合单元节点和单元中心函数值计算三阶空间导数，通过指标半平移避免传递误差，具有高阶、高分辨率和低耗散特性，适用于含三阶导数项的色散波方程。
摘要：
本文受现有单元节点型和单元中心型紧致差分格式的启发，提出一类新的中心紧致格式，该格式在求解色散波方程时具有更优的谱分辨率。在我们的方法中，同时利用单元节点处和单元中心处的函数值来计算单元节点处的三阶空间导数。为了计算单元中心处的空间导数，我们采用一种技术，即在最初为单元节点设计的公式中对指标进行半平移。与传统的紧致插值格式相比，所提方法有效避免了传递误差的引入。我们采用基于泰勒级数展开的方法计算有限差分系数。通过系统的傅里叶分析和数值试验，我们注意到这些方法表现出高阶、高分辨率和低耗散等优异特性。计算结果进一步说明了高阶紧致格式的有效性，尤其是在处理含三阶导数项的问题时。
英文摘要：
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.
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