The Application of Model Order Reduction Methods to a Linear Cross-Diffusion Model

This work investigates reduced-order modelling (ROM) strategies for the Brusselator system, a linear cross-diffusion framework with applications in chemical and biological systems. Spatial discretization is performed via the symmetric interior penalty discontinuous Galerkin (SIPG) method, and time-stepping is handled via the backward Euler scheme to derive the full-order model (FOM). To construct reduced-order bases, singular value decomposition (SVD) is applied to a snapshot matrix of FOM solutions, followed by proper orthogonal decomposition (POD) to generate low-dimensional approximations. However, the nonlinear terms in the ROM retain the original high-dimensional structure of the FOM. To mitigate this computational burden, the discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) are integrated to approximate nonlinearities efficiently. Numerical experiments conducted on a 2D Brusselator system validate the accuracy and efficiency of the proposed POD-DEIM and POD-DMD frameworks, with both methods achieving close agreement with the FOM.