Comparison of dissipative-dispersion properties of some conservative difference schemes

Schemes with a compact template are very attractive for the numerical solution of the transport equation in the presence of complex contact discontinuities and external boundaries. A numerical scheme in which the construction is based on a minimal two-point stensil in each direction is called bicompact. Without expanding the list of required values in the cell, the maximum order of approximation of the scheme is two. The class of such schemes includes the Goloviznin–Chetverushkin scheme. To increase the approximation order, it is needed to expand the list of required variables. The approximation orders can be independent in space and time, both in bicompact Rogov schemes, and consistent, as is most often the case in interpolation-characteristic methods. In this paper, the dissipative-dispersion properties of three different conservative bicompact schemes for advection equation are investigated. It is shown that a modification of the CIP (Cubic Interpolation Polynomial) scheme based on Hermitian interpolation has an extra-small variance for almost any Courant numbers, which makes it very weakly non-monotonic.