单个守恒型方程熵耗散格式中熵耗散函数的构造

对于一维单个守恒律方程,文8]设计了一种非线性守恒型差分格式.此格式为二阶Godunov型的,用的是分片线性重构(reconstruction),重构函数的斜率是根据熵耗散得到的.格式满足熵条件.与传统的守恒格式不同的是此格式在计算过程中不仅用到了数值解还用到了数值熵.在此格式中一个所谓的熵耗散函数起到了很重要的作用,它在每一个网格的计算中耗散熵,以保证格式满足熵条件.文8]中设计的熵耗散函数比较复杂,并且不是很完善.故数值地分析了在格式的构造中为何应给熵以一定的耗散,及应耗散多少.并且给出了一个新的以数值解的二阶差分作为基本模块的熵耗散函数.最后给出了相应的数值算例.

The Design of the Entropy Dissipator of the Entropy Dissipating Scheme for Scalar Conservation Law

In our foregoing paper 8] we had designed a nonlinear conservative difference scheme of second-order Godunov type with piecewise-linear reconstruction,in which the slope of the reconstructed function in each grid cell can be computed by dissipating the entropy. Such a scheme satisfies the entropy condition,and computes not only numerical solution but also numerical entropy; thus, it is different from all former conservative schemes. A socalled entropy dissipator in the scheme, which dissipates the entropy in each grid cell in the computation, plays an important role in stabilizing the computation. The entropy dissipator designed in is quite complicated. In this paper, we numerically discuss why entropy must be dissipated and how much entropy should be dissipated. A new entropy dissipator, based on the second-order difference of the numerical solution is given. Numerical examples are presented to show how the entropy dissipator suppresses nonphysical oscillations near discontinuities.