Entropy Dissipating Scheme For Hyperbolic Systems Of Conservation Laws In One Space Dimension

In this paper, we are concerned with hyperbolic conservation laws in one space dimension. We describe a method to design high resolution schemes for the equations. The designed scheme is of Godunov-type with piecewise-line reconstruction. Different from all other Godunov-type schemes for the conservation laws, our scheme computes not only the numerical solution but also an approximation to the entropy, called numerical entropy. In the scheme the reconstruction of solution is performed by requiring the cell-average of the entropy of the reconstructed solution to be equal to the numerical entropy in each grid cell. Both the numerical solution and numerical entropy are computed in a finite-volume fashion while the computation of the latter involves a so-called entropy dissipation term, which simulates the variation of the entropy. In doing so, the numerical dissipation is introduced in the scheme to stabilize the computation.The definition of consistency of the schemes is given and a Lax-Wendroff theory for the scheme is also given, which says that if the numerical solution converges, it converges to an entropy solution of the orginal equation.Since a major motivation of designing this kind of scheme is to try to overcome the numerical dissipation in the linearly degenerated characteristic field in the system case, we conduct a numerical study of the scheme without entropy dissipation term on the linear advection equation to investigate the way in which the numerical entropy eliminates the linear dissipation. Based on this study, we designed two schemes of the type. In one of the schemes, certain entropy dissipation term is still involved in the computation of the entropy in the linearly degenerated characteristic field; thus, the numerical entropy is not coservative there. In the other scheme, we set the entropy dissipation in the linear field to be zero so that the numerical entropy is conservative in this field. We then design a so-called "Minimums-Increase-and-Maximums-Decrease" (MIMD) slope-limiter in the reconstruction step of the scheme to eliminate the non-physical oscillation then caused.