Volume 24, Issue 6

The existence and uniqueness theorems ‘in the large’ are proved for a system of primitive equations in the Cartesian coordinates in a domain with an uneven bottom. The original equations are slightly modified: some terms containing mixed derivatives are omitted because they are small. Namely, it is proved that for arbitrary time period [0, T ] in a spatial domain Ω = {( x,y,z ) | ( x,y ) ∈ Ω′, z ∈ [0, H ( x,y )]}, for an arbitrary viscosity coefficients ν,ν 1 > 0, any depth H ∈ C 2 (Ω′), H ⩾ H 0 > 0, and any initial conditions there exists a unique weak solution and the norms are continuous with respect to t , where s is the vertical variable.

A finite difference scheme on semi-staggered grids is used for numerical simulation of unsteady flows of an incompressible viscoplastic Bingham medium. The Duvaut–Lions variational inequality is considered as a mathematical model. The convergence of the Uzawa method is proved. The efficiency of the method is demonstrated on a model problem with a known exact solution (plane Poiseuille flow) and on the unsteady driven cavity problem.

Numerical methods and studies of differential properties of an R ν -weak solution are presented for boundary value problems with a divergent Dirichlet integral of the solution.

An initial boundary value problem for a singular perturbed parabolic reaction–diffusion equation is considered in a domain unbounded in x on the real axis; the leading derivative of the equation contains the parameter ε 2 ; ε ∈ (0, 1]. The right-hand side of the equation and the initial function indefinitely grow as 𝒪( x 2 ) for x → ∞, which leads to an indefinite growth of the solution at infinity as 𝒪(Ψ( x )), where Ψ( x ) = x 2 + 1. For small values of the parameter ε a parabolic boundary layer appears in the neighbourhood of the lateral part of the boundary. In this problem for fixed values of the parameter ε the error of the grid solution indefinitely grows in the uniform norm for x → ∞. The closeness of the solutions to the initial boundary value problem and to its grid approximations is considered in this paper in the weighted uniform norm with the weight function Ψ –1 ( x ); the solution to the initial boundary value problem is ε -uniformly bounded in this norm. Using special grids condensing in the neighbourhood of the boundary layer, we construct formal difference schemes (schemes on grids with an infinite number of nodes) convergent ε -uniformly in the weighted norm. Based on the notion of the domain of essential dependence of the solution, we propose constructive difference schemes (schemes on grids with a finite number of nodes) convergent ε -uniformly in the weighted norm on given bounded subdomains.