Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation^∗

¹ Department of Mathematics and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China.
sfzhu@math.ecnu.edu.cn
² CMCS-Chair of Modeling and Scientific Computing, MATHICSE-Mathematics Institute of Computational Science and Engineering, EPFL-École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland.
luca.dede@epfl.ch; alfio.quarteroni@epfl.ch
³ MOX-Modeling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy.

Received: 26 January 2016
Revised: 19 June 2016
Accepted: 26 August 2016

Abstract

Discretization methods such as finite differences or finite elements were usually employed to provide high fidelity solution approximations for reduced order modeling of parameterized partial differential equations. In this paper, a novel discretization technique-Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations. We propose a new fully discrete IGA-Newmark-POD approximation and we analyze the associated numerical error, which features three components due to spatial discretization by IGA, time discretization with the Newmark scheme, and modes truncation by POD. We prove stability and convergence. Numerical examples are presented to show the effectiveness and accuracy of IGA-based POD techniques for the model order reduction of the acoustic wave equation.