A reduced-order extrapolating collocation spectral method based on POD for the 2D Sobolev equations

Shiju Jin,Zhendong Luo

doi:10.1186/s13661-019-1176-2

Shiju Jin, Zhendong Luo

Open Access

https://doi.org/10.1186/s13661-019-1176-2

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Journal: Boundary Value Problems	Publication Date: Mar 28, 2019
Citations: 7	License type: open-access

Affiliation: North China Electric Power University

Abstract

In this paper, we mainly use a proper orthogonal decomposition (POD) to reduce the order of the coefficient vectors of the solutions for the classical collocation spectral (CS) method of two-dimensional (2D) Sobolev equations. We first establish a reduced-order extrapolating collocation spectral (ROECS) method for 2D Sobolev equations so that the ROECS method includes the same basis functions as the classic CS method and the superiority of the classic CS method. Then we use the matrix means to discuss the existence, stability, and convergence of the ROECS solutions so that the procedure of theoretical analysis becomes very concise. Lastly, we present two set of numerical examples to validate the effectiveness of theoretical conclusions and to illuminate that the ROECS method is far superior to the classic CS method, which shows that the ROECS method is quite valid to solve Sobolev equations. Therefore, both theory and method of this paper are completely different from the existing reduced-order methods.

Highlights

Let Ω ⊂ R2 be an open bounded domain with boundary ∂Ω
4 Some numerical examples we present several sets of comparative numerical examples to show the advantage of the reduced-order extrapolating collocation spectral (ROECS) method for the 2D Sobolev equation
5 Conclusions and discussions In this study, we have studied the reduced-order of the coefficient vectors of the solutions for the classic collocation spectral (CS) method of 2D Sobolev equations

Summary

Introduction

Let Ω ⊂ R2 be an open bounded domain with boundary ∂Ω. We consider the following two-dimensional (2D) Sobolev equation:. Though the classic CS method (see [16]) for 2D Sobolev equations can attain higher accuracy than the FD scheme, FE method, and FVE method, it contains a lot of degrees of freedom (unknowns). In this way, because of the round-off error accumulation in numerical calculations, after several computational steps, there generally occurs a floating-point overflow such that we cannot obtain the desired consequences. In this paper, utilizing POD to reduce the order of the coefficient vectors of the CS solutions for the classic CS method, we construct a ROECS method only holding few degrees of freedom.

The variational formulation for the 2D Sobolev equations

The matrix representation of the classic CS format

Formulation of POD basis

Establishment of the ROECS model

The flowchart for solving the ROECS format

Conclusions and discussions