Extrapolation Algorithms Of Numerical Solutions Based On POD Method For Parabolic Equations And Viscoelastic Equation

The article mainly gives some reduced-order extrapolation difference algorithms based on proper orthogonal decomposition (POD) method for the Sobolev equation and viscoelastic equation. In order to avoid the disadvantage of finite difference (FD) schemes with large amount of calculation for Sobolev Equation and viscoelastic equation, we employ the singular value decomposition and POD method to reduce the classical explicit FD scheme and the C-N FD scheme for the Sobolev equation and for the FD scheme for the viscoelastic equation, and build some new reduced-order extrapolation FD schemes. The new reduced-order extrapolation FD schemes use very little initial data values to establish some POD-based reduced-order extrapolation FD schemes with very few degrees of freedom, thus saving the computer’s CPU and memory and reducing the accumulation of truncation error in the computing process so that solving numerical solutions becomes more effective.This article is taken step by step strategy. First, we explore the classical explicit FD Scheme for the Sobolev equation and the corresponding reduced-order extrapolation FD scheme that are the condition stable with fisrt-order time accuracy. Then, we further describes the C-N FD scheme and the corresponding reduced-order extrapolation FD scheme that are the unconditionally stable implicit FD scheme and have full secone-order accuracy. Another important work in this paper is to explore the classical FD scheme for the viscoelastic equations and the corresponding POD-based reduced-order FD extrapolation scheme. Because the viscoelastic equation includes the secome-order time derivative, it is different from the Sobolev equation and is far more complex than the Sobolev equation, but is more useful than the Sobolev equation. In this article, we provide the error estimates the extrapolation reduced-order FD schemes above mentioned and gives the implementation procedure of the extrapolation FD schemes. Finally, we provide some examples to verify the correctness and validity of these POD-based methods. It is sufficiently shown that the POD-based extrapolation reduced-order FD schemes have to be reckoned with application value and immeasurable potential.