Regularized shallow water equations and a new method of numerical simulation of the open channel flows

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

“…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.…”

On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

“…Notice that due to Formula (15), we have κ 2 0 (θ) − a 0 θ 2 > 0 excluding the particular case θ = −1 = − a 0 . From ( 14) and (15), it also follows that κ(θ) increases in θ 0.…”

“…These methods are widely used for numerical solution of various applied problems. In the barotropic case, gas dynamics systems of equations with such regularizations were introduced and studied in [10][11][12][13], and their numerous applications to computer simulation were given for various 1D and 2D shallow water models [14][15][16][17][18][19], some 2D astrophysical problems [20] and the 2D and 3D compressible Navier-Stokes-Cahn-Hilliard models [21][22][23], etc. Despite of a lot of applications, almost nothing was known until recent years on rigorous theoretical conditions for stability of schemes with the QGD and QHD regularizations thus leading to additional time-consuming preliminary numerical experiments in order to choose the adequate parameters of schemes allowing to use larger values of ∆t.…”