The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method.

This article considers mixed problem for 2-dimensional symmetric t-hyperbolic system. This system consists of constant coefficients. We investigated questions of setting mixed problems (problems with initial-boundary conditions) for symmetric t-hyperbolic system. It is known that for a symmetric t-hyperbolic system, the concept of characteristic velocity plays a very important role. The main point of this article is to construct a difference control scheme for these characteristic velocities. Control parameters are introduced to control the characteristic velocities of a symmetric t-hyperbolic system. The value of the parameter, the characteristic velocities can have different signs. We will consider all possible cases. In these cases, it is obvious that a symmetric t-hyperbolic system with n-equations can be in (n+1)(n+1) different states. These states controlled by control parameters. Upwind difference schemes constructed for each of these states depending on the value of characteristic velocities. The suggested difference schemes for the mixed problem were substantiated is exponential stability by the Lyapunov. It will be shown that the difference scheme will be stable if the CFL condition is satisfied and otherwise unstable.

A new linear model of unsteady flow in a real pump (compressor station) – pressure pipeline – valve (pressure regulator) system is constructed. The model is a symmetric t-hyperbolic system that accounts for friction. The boundary conditions are expressed through the slope of the pump characteristic s = dH/dQ and the linearised valve impedance ζ, and they reduce to the dimensionless reflection coefficients R₁ = −(1+σ)/(1−σ) and R_L = (β−1)/(β+1), where σ = s/Z, β = ζ/Z and Z = a/(gA) is the wave impedance of the pipe. Without friction, the system is shown to be exponentially stable if and only if (σ−β)(σ−β⁻¹) > 0; with friction, the characteristic equation is derived. An upwind difference scheme is proposed in which the diagonal part of the friction term is treated implicitly and the coupling part explicitly. The scheme is solved by explicit sweep formulas without matrix inversion. Its L₂-stability and, by means of a weighted discrete analogue of the Lyapunov function, its exponential stability are proved. In numerical experiments, the water hammer caused by partial valve closure agrees with the Joukowsky formula to within 1.5%. Oscillations decay in normal operation, whereas self-excited oscillations arise for a throttled valve on the rising branch of the pump characteristic. The numerical stability boundaries coincide with the theoretical boundaries obtained from the characteristic equation.