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Shallow-water equations in a domain representing a strait are transformed into curvilinear coordinates concordant with the domain configuration and mapping the domain onto a rectangle with two opposite open boundaries. The initial boundary value problem for the equations in the form of contravariant fluxes is considered and compared with a reduced formulation of boundary conditions for equations linearized near the boundary. A method for numerical integration of the transformed boundary value problem is discussed. Particular attention is given to modification ensuring stable computation in the case of strong non-linearity and variability of domain metric.