Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

• Published in: International journal for numerical methods in fluids 2014-11, Vol.76 (8), p.497-521
• Main Authors: Ştefănescu, Răzvan, Sandu, Adrian, Navon, Ionel M.
• Format: Article
• Language: English
• Subjects: alternating direction implicit methods (ADI); Calculus; Complexity; discrete empirical interpolation method (DEIM); finite difference methods; Mathematical analysis; Mathematical models; Nonlinearity; Online; Polynomials; reduced order models (ROMs); shallow water equations (SWE); tensorial proper orthogonal decomposition (POD); Two dimensional
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.3946

SUMMARYThis paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of O(kp+1), where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two‐dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76× the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8× slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. Copyright © 2014 John Wiley & Sons, Ltd. This paper introduces tensorial calculus techniques in the framework of proper orthogonal decomposition (POD) to reduce the computational complexity of the reduced polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of O(kp + 1) ‐ order of k at the power p+1, where k is the dimension of POD basis and therefore is independent of full space dimension. Reduced two‐dimensional shallow water equation models based on tensorial POD, POD, and POD/discrete empirical interpolation method are developed and their performances are extensively analyzed.