Construction of asymptotics of a discrete solution based on nonclassical differential approximations

A method is proposed that yields an asymptotic expansion of a discrete solution to the Riemann problem. The method is based on the concept of the determining coefficient of an asymptotic expansion, which is used to construct a nonclassical differential approximation to a finite-difference scheme. The method is described by using linear finite-difference schemes approximating the linear advection equation. Asymptotic expansions of a discrete solution are constructed for explicit two-level schemes with artificial viscosity and dispersion and for a symmetric compact finite-difference scheme with second- and fourth-order conservative artificial viscosities. It is shown that the structure of the discrete solution on a shock front is fairly accurately described by the expansions constructed.