POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

Abstract

In this work we report on a reduced-order model (ROM) for the system of time-domain Maxwell's equations discretized by a discontinuous Galerkin (DG) method. We leverage previous results on proper orthogonal decomposition (POD) [1,2], in particular for the wave equation [3], to propose a POD-based ROM with an adaptive snapshot selection strategy where the snapshots are produced by a high order discontinuous Galerkin time-domain (DGTD) solver. The latter is formulated on an unstructured simplicial mesh, and combines a centered scheme for the definition of the numerical fluxes of the electric and magnetic fields at element interfaces with a second order leap-frog (LF2) time scheme for the time integration of the associated semi-discrete equations. The POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto a low-dimensional space generated by the POD basis vectors. Inspired from the approach followed in [2,3], we derive error bounds for the POD-based ROM that is adapted to our particular modeling and discretization settings. The adaptive snapshot selection algorithm exploits the results of this analysis to measure the control error. A snapshot choosing rule aiming at keeping the error estimate close to a target selection error tolerance is proposed, which is similar to the standard rules found in adaptive time-stepping ordinary differential equations (ODEs) solvers. An incremental singular value decomposition (ISVD) algorithm is used to update the SVD on-the-fly when a new snapshot is available. The purpose of this adaptive selection strategy is to save memory without storing snapshots, while producing a smaller error. Numerical experiments for the 2-D time-domain Maxwell's equations nicely illustrate the performance of the resulting POD-based ROM with adaptive snapshot selection.