Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

Abstract

This paper is concerned with establishing a reduced-order extrapolating finite volume element (FVE) format based on proper orthogonal decomposition (POD) for two-dimensional (2D) hyperbolic equations. For this purpose, a semi discrete variational format relative time and a fully discrete FVE format for the 2D hyperbolic equations are built, and a set of snapshots from the very few FVE solutions are extracted on the first very short time interval. Then, the POD basis from the snapshots is formulated, and the reduced-order POD extrapolating FVE format containing very few degrees of freedom but holding sufficiently high accuracy is built. Next, the error estimates of the reduced-order solutions and the algorithm procedure for solving the reduced-order format are furnished. Finally, a numerical example is shown to confirm the correctness of theoretical conclusions. This means that the format is efficient and feasible to solve the 2D hyperbolic equations.