In this paper, an oscillation-free spectral volume (OFSV) method is proposed and studied for the hyperbolic conservation laws. The numerical scheme is designed by introducing a damping term in the standard spectral volume method for the purpose of controlling spurious oscillations near discontinuities. Based on the construction of control volumes (CVs), two classes of OFSV schemes are presented. A mathematical proof is provided to show that the proposed OFSV is stable and has optimal convergence rate and some desired superconvergence properties when applied to the linear scalar equations. Both analysis and numerical experiments indicate that the damping term would not destroy the order of accuracy of the original SV scheme and can control the oscillations discontinuities effectively. Numerical experiments are presented to demonstrate the accuracy and robustness of our scheme.

翻译：本文提出并研究了一种无振荡谱体积（OFSV）方法，用于求解双曲守恒律。该数值格式通过在标准谱体积方法中引入阻尼项来设计，旨在控制间断附近产生的虚假振荡。基于控制体积（CV）的构造，本文提出了两类OFSV格式。数学证明表明，所提出的OFSV方法具有稳定性，且在应用于线性标量方程时具有最优收敛速率及一些期望的超收敛特性。分析和数值实验均表明，阻尼项不会破坏原始SV格式的精度阶，并能有效控制间断处的振荡。数值实验展示了该格式的精度和稳健性。