A POD-based ROM strategy for the prediction in time of advection-dominated problems

Abstract

The use of reduced-order models (ROMs) for the numerical approximation of the solution of partial differential equations is a topic of current interest, being motivated by the high computational efficiency of ROMs when compared to full-order models (FOMs). To construct a ROM to approximate the solution of transport equations, the use of the proper orthogonal decomposition (POD) method is a common choice. POD-based ROMs rely on the snapshot method, which consists in the off-line computation of a set of values corresponding to the solution up to the training time by means of the FOM. Then, the ROM is constructed and solved, up to the training time. When considering parabolic equations, the method is able to compute the solution beyond the training time. However, when considering hyperbolic problems, POD-based ROMs fail when computing the solution beyond the training time, this being one of the strongest limitations of POD-based ROMs. In this work, a strategy in the framework of POD-based ROMs to predict solutions in time is introduced. This method, called CT-ROM, is based on a coordinate transformation and allows to compute the solution of advection-dominated problems beyond the training time. The performance of this strategy is assessed using a variety of test cases, showing promising results in all of them. The extension of the CT-ROM to higher spatial dimensions by means of the Radon transform is also presented. The results obtained are encouraging and motivate the application of this idea to more complex problems.