A Reduced-Order Extrapolation Spectral-Finite Difference Scheme Based on the POD Method for 2D Second-Order Hyperbolic Equations

Abstract

In this study, a reduced-order extrapolation spectral-finite difference (ROESFD) scheme based on the proper orthogonal decomposition (POD) method is set up for the two-dimensional (2D) second-order hyperbolic equations. First, the classical spectral-finite difference (CSFD) method for the 2D second-order hyperbolic equations and its stability, convergence, and flaw are introduced. Then, a new ROESFD scheme that has very few degrees of freedom but holds sufficiently high accuracy is set up by the POD method and its implementation is offered. Finally, three numerical examples are offered to explain the validity of the theoretical conclusion. This implies that the ROESFD scheme is viable and efficient for searching the numerical solutions of the 2D second-order hyperbolic equations.

Keywords:

• reduced-order extrapolation spectral-finite difference scheme
• proper orthogonal decomposition
• second-order hyperbolic equations
• classical spectral-finite difference method
• error estimates

AMS Subject Classification:

• 65N12
• 65M15
• 65N30