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

In the present paper we consider a 2-D shallow-water equations (SWE) model on a βplane solved using an alternating direction fully implicit (ADI) finite-difference scheme (Gustafsson 1971, Fairweather and Navon 1980, Navon and De Villiers 1986, Kreiss and Widlund 1966) on a rectangular domain. The scheme was shown to be unconditionally stable for the linearized equations. The discretization yields a number of nonlinear systems of algebraic equations. Then we use a proper orthogonal decomposition to reduce the dimension of the SWE model. Due to the model nonlinearities, the computational complexity of the reduced model still depends on the number of variables of the full shallow water equations model. By employing the discrete empirical interpolation method (DEIM) we reduce the computational complexity and regain the full model reduction expected from the POD model. To emphasize the performances of DEIM, we also propose an explicit Euler finite difference scheme (EE) as an alternative to ADI scheme for solving the swallow water equations model. We assess the efficiency of DEIM as a function of number of spatial discretized points, snapshots, and POD basis functions. The CPU time was decreased by a factor of 10/15 in case of DEIM implicit/explicit SWE scheme when the number of spatial discretized ∗Corresponding author Preprint submitted to Elsevier June 8, 2012 points was more than 10000. More, once the number of points selected by DEIM algorithm reached 40, the approximation errors from POD/DEIM and POD reduced systems are indistinguishable. We also studied the RMSE errors and the correlation coefficients between full model, POD and POD/DEIM reduced systems as well as the conservation of the integral invariants of the SWE model. 1−day experiments showed that the POD/DEIM ADI SWE scheme conserves the integral invariants of the SWE model.