We develop in this paper a fifth-order nonlinear spectral difference method for solving hyperbolic conservation laws, whose solutions often admit discontinuities. To avoid instability caused by the Gibbs phenomenon arising from interpolation across discontinuities, a fifth-order nonlinear interpolation scheme is proposed within a single cell, keeping the compactness of the original linear spectral difference method. Some numerical results are also presented to demonstrate the accuracy and effectiveness of the proposed method.

中文翻译：

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双曲守恒律的五阶非线性谱差分格式

在本文中，我们开发了一种用于求解双曲守恒定律的五阶非线性谱差方法，该方法的解常允许不连续性。为了避免由不连续点之间的插值引起的吉布斯现象引起的不稳定性，在单个像元中提出了一种五阶非线性插值方案，同时保持了原始线性谱差方法的紧凑性。一些数值结果也被提出来证明所提方法的准确性和有效性。