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

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 O(1000).