It is well known that the conventional shock capturing schemes for hyperbolic systems of conservation laws yield oscillatory solutions near discontinuities when the mesh is not fine enough. In this article, an improved finite volume scheme (IFVS) is proposed to simulate hyperbolic conservation laws. In this scheme, some weighting coefficients are introduced to discretize the convection term. By eliminating the truncation error of the numerical scheme, the weighting coefficient and the ratio of space step to time step are determined analytically. Theoretical and numerical results show that although the same computational stencil as the classical finite volume method (FVM) is employed to divide the computational domain and the simple explicit forward difference is used to discretize the time derivative, the proposed IFVS can achieve almost the same accuracy as the exact solution for the scalar hyperbolic conservation laws. Furthermore, for the hyperbolic conservation laws involving shocks, the numerical solutions obtained by the IFVS do not show any unphysical oscillations even a very coarse mesh is utilized to divide the computational domain, which is consistent with the theoretical analysis showing that the IFVS is unconditionally stable.

中文翻译：

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双曲守恒律的一种新的有限体积方案

众所周知，当网格不够细时，用于守恒定律的双曲线系统的常规冲击捕获方案会在不连续点附近产生振荡解。在本文中，提出了一种改进的有限体积方案（IFVS）来模拟双曲守恒律。在该方案中，引入一些加权系数以离散对流项。通过消除数值方案的截断误差，可以解析地确定加权系数和空间步长与时间步长之比。理论和数值结果表明，尽管采用与经典有限体积法（FVM）相同的计算模板来划分计算域，并使用简单的显式正向差异来离散时间导数，提出的IFVS可以达到与标量双曲守恒定律的精确解几乎相同的精度。此外，对于涉及冲击的双曲守恒律，IFVS所获得的数值解即使使用非常粗糙的网格划分计算域也不会显示任何非物理振动，这与理论分析一致，表明IFVS是无条件稳定的。