Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media

Abstract

A discrete empirical interpolation method (DEIM) is applied in conjunction with proper orthogonal decomposition (POD) to construct a non-linear reduced-order model of a finite difference discretized system used in the simulation of non-linear miscible viscous fingering in a 2-D porous medium. POD is first applied to extract a low-dimensional basis that optimally captures the dominant characteristics of the system trajectory. This basis is then used in a Galerkin projection scheme to construct a reduced-order system. DEIM is then applied to greatly improve the efficiency in computing the projected non-linear terms in the POD reduced system. DEIM achieves a complexity reduction of the non-linearities, which is proportional to the number of reduced variables, whereas POD retains a complexity proportional to the original number of variables. Numerical results demonstrate that the dynamics of the viscous fingering in the full-order system of dimension 15,000 can be captured accurately by the POD–DEIM reduced system of dimension 40 with the computational time reduced by factor of .

Keywords:

• Non-linear model reduction
• proper orthogonal decomposition
• empirical interpolation methods
• non-linear partial differential equations
• miscible viscous fingering

Acknowledgements

We thank Prof. Beatrice Riviere for suggesting this miscible flow problem, and for giving helpful advice and comments throughout the course of this work. This work was supported in part by AFOSR grant FA9550-09-1-0225 and by NSF grant DMS-0914021.