Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
Computers & Mathematics with Applications
Nonlinear Model Reduction via Discrete Empirical Interpolation
SIAM Journal on Scientific Computing
Proceedings of the 48th Design Automation Conference
Numerical simulations with data assimilation using an adaptive POD procedure
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction
SIAM Journal on Numerical Analysis
A numerical investigation of velocity-pressure reduced order models for incompressible flows
Journal of Computational Physics
Non-linear model reduction for the Navier-Stokes equations using residual DEIM method
Journal of Computational Physics
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In the present paper we consider a 2-D shallow-water equations (SWE) model on a @b-plane solved using an alternating direction fully implicit (ADI) finite-difference scheme 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. We then use a proper orthogonal decomposition (POD) 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 of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the POD model. To emphasize the CPU gain in performance due to use of POD/DEIM, we also propose testing an explicit Euler finite difference scheme (EE) as an alternative to the ADI implicit scheme for solving the swallow water equations model. We then proceed to assess the efficiency of POD/DEIM as a function of number of spatial discretization points, time steps, and POD basis functions. As was expected, our numerical experiments showed that the CPU time performances of POD/DEIM schemes are proportional to the number of mesh points. Once the number of spatial discretization points exceeded 10000 and for 90 DEIM interpolation points, the CPU time decreased by a factor of 10 in case of POD/DEIM implicit SWE scheme and by a factor of 15 for the POD/DEIM explicit SWE scheme in comparison with the corresponding POD SWE schemes. Moreover, our numerical tests revealed 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 same orders of magnitude, thus supporting the theoretical results existing in the literature.