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In the present paper we consider a 2-D shallow-water equations (SWE) model ona $\beta$-plane solved using an alternating direction fully implicit (ADI)finite-difference scheme on a rectangular domain. The scheme was shown to beunconditionally stable for the linearized equations.  The discretization yields a number of nonlinear systems of algebraicequations. We then use a proper orthogonal decomposition (POD) to reduce thedimension of the SWE model. Due to the model nonlinearities, the computationalcomplexity of the reduced model still depends on the number of variables of thefull shallow - water equations model. By employing the discrete empiricalinterpolation method (DEIM) we reduce the computational complexity of thereduced order model due to its depending on the nonlinear full dimension modeland regain the full model reduction expected from the POD model.  To emphasize the CPU gain in performance due to use of POD/DEIM, we alsopropose testing an explicit Euler finite difference scheme (EE) as analternative to the ADI implicit scheme for solving the swallow water equationsmodel.  We then proceed to assess the efficiency of POD/DEIM as a function of numberof spatial discretization points, time steps, and POD basis functions. As wasexpected, our numerical experiments showed that the CPU time performances ofPOD/DEIM schemes are proportional to the number of mesh points. Once the numberof spatial discretization points exceeded 10000 and for 90 DEIM interpolationpoints, the CPU time was decreased by a factor of 10 in case of POD/DEIMimplicit SWE scheme and by a factor of 15 for the POD/DEIM explicit SWE schemein comparison with the corresponding POD SWE schemes. Our numerical testsrevealed that if the number of points selected by DEIM algorithm reached 50,the approximation errors due to POD/DEIM and POD reduced systems have the sameorders of magnitude.

POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model