Comparison of dissipation and dispersion properties of compact difference schemes for the numerical solution of the advection equation

The dissipation and dispersion properties of the Hermite characteristic scheme intended for solving the one-dimensional advection equation are examined. The scheme is based on Hermite interpolation using not only the nodal values of the function, but also the nodal values of its spatial derivative. The derivatives at a new time step are computed so as to ensure the correct redistribution of the input fluxes over the output faces. Note that the scheme is constructed within a single cell, so it belongs to the class of bicompact schemes. The derivatives at a new time level are reconstructed using an integral average and the Euler–Maclaurin formula. The scheme is compared with modern conservative schemes, such as the bicompact Rogov scheme and the Goloviznin–Chetverushkin scheme. It is shown that the Hermite characteristic scheme has low dissipation and ultralow dispersion as compared with schemes of the same class. The dispersion of the Hermite characteristic scheme is lower than that of the semidiscrete bicompact Rogov scheme. In turn, the latter scheme with time approximation based on the trapezoidal rule has zero dissipation. Similar ideas of using characteristic schemes with an additional algorithm ensuring conservativeness are used in the Goloviznin–Chetverushkin scheme, which is the simplest in implementation. The schemes used for comparison have a compact stencil and similar technics used for closing difference schemes.