On a Conservative Finite-Difference Method for 1D Shallow Water Flows Based on Regularized Equations

On Conditions for Weak Conservativeness of Regularized Explicit Finite-Difference Schemes for 1D Barotropic Gas Dynamics Equations

“…is known to be a necessary condition for property (21) to hold (see similar result in [6]). Hereafter λ l (A) are eigenvalues of a matrix A.…”

“…Hereafter λ l (A) are eigenvalues of a matrix A. Let us determine the spectral form of property (21).…”

“…It is essential to notice that for the linearized QGD-system (2)-( 4) namely the corresponding inequality holds in place of equality (7) so that it is natural to study the bound for schemes based on such a system. Of course, estimate (21) guarantees the uniform in time stability in H with respect to initial data. We first substitute a partial solution in the form y m k = e ikξ v m (ξ ), k ∈ Z, m ≥ 0, where i is the imaginary unit and 0 ≤ ξ ≤ 2π is a parameter, into (20) and obtain…”

“…Remark 1. Under validity of the spectral bound (25), the norm y m H is actually non-increasing in m ≥ 0 that serves as a stronger property than (21).…”

“…Their construction is based on special quasi-gasdynamic and quasihydrodynamic [3,8,13,15,20] regularizations of the original equations (without a regularization, the schemes are unstable). The schemes of this kind were suc- cessfully applied in numerous and various practical applications, in particular, see [2,9,10,21], but their theory is not developed so well.…”