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

The reduced-order extrapolation difference algorithms based on proper orthogonal decomposition (POD) method for parabolic equations and the Sobolev equations are mainly discussed in this article. In order to overcome the deficiency about large calculation of the difference scheme for parabolic equations that is not suitable for large-scale computation, singular value decomposition technique and POD method are employed to reduce the dimension of the classical finite difference scheme and Crank-Nicolson scheme for parabolic equations respectively, then establish new extrapolation difference schemes. With few known first staps solutions, a group of reduced-order difference scheme with few free degrees could build by means of the POD bases, which could save a great deal of the computer memory and reduce the calculation time. Due to reducing the large scale of computational load, this reduced-order extrapolation difference schemes can lessen the accumulation of truncation error in the computational process, so that the numerical solutions of the equation are higher accuracy.The process of this paper is from the shallow to the deep, the classical difference scheme and its reduced-order extrapolation algorithm for the parabolic equations are discussed first, which are explicit scheme, conditional stability and the first-order time accuracy. And then the Crank-Nicolson format and its reduced-order extrapolation algorithm are presented, which are implicit scheme, unconditional stability and fully second-order accuracy. Finally, the classical difference method and reduced-order extrapolation difference algorithm for the Sobolev equations are established, which are similar to forms for parabolic equations. Since Sobolev equations include more of a high order derivative term and its research method and content are similar to parabolic equations, we explore them together. Mean while, the ideology of extrapolation algorithm has successfully applied to computing the difference scheme for Sobolev equations, a good application effect is attained. Through strict deduction and proof, the error estimates of these reduced-order extrapolation difference schemes are obtained respectively. The implementation steps of these extrapolation algorithms are given respectively. The theoretical results are all verified with numerical experiments. The application value of these reduced-order extrapolation difference algorithms is illustrated adequately.