A reduced fourth-order compact difference scheme based on a proper orthogonal decomposition technique for parabolic equations

Baozou Xu,Xiaohua Zhang

doi:10.1186/s13661-019-1243-8

Abstract

In this paper, we present a reduced high-order compact finite difference scheme for numerical solution of the parabolic equations. CFDS4 is applied to attain high accuracy for numerical solution of parabolic equations, but its computational efficiency still needs to be improved. Our approach combines CFDS4 with proper orthogonal decomposition (POD) technique to improve the computational efficiency of the CFDS4. The validation of the proposed method is demonstrated by four test problems. The numerical solutions are compared with the exact solutions and the solutions obtained by the CFDS4. Compared with CFDS4, it is shown that our method has greatly improved the computational efficiency without a significant loss in accuracy for solving parabolic equations.

Highlights

Many problems in physical phenomena, engineering equipment, and living organisms, such as the proliferation of gas, the infiltration of liquid, the conduction of heat, and the spread of impurities in semiconductor materials, can be described with parabolic equations [1,2,3]
We first obtain the optimal basis from the snapshot and use the optimal basis to derive a R-CFDS4 for parabolic equations
We fix h as 0.0025, the maximal error is reduced by a factor about 22 each time in Table 10, which indicates that the R-CFDS4 is second-order accurate in time

Summary

Introduction

Many problems in physical phenomena, engineering equipment, and living organisms, such as the proliferation of gas, the infiltration of liquid, the conduction of heat, and the spread of impurities in semiconductor materials, can be described with parabolic equations [1,2,3]. The high-order compact finite difference scheme (CFDS) has been implemented for numerical solution of various types of partial differential equations. Such as the Rosenau-regularized long wave (RLW) equation [11], integro-differential equations [12], Burgers’ equation [13], Helmholtz equation [14], Navier–Stokes equations [15], Schrödinger equations [16], Poisson equation [17], sine-Gordon equation [18]. The main goal of this paper is to construct a numerical algorithm which has high computational accuracy and efficiency for solving parabolic equations.

The fourth-order compact FDS for 1D parabolic equation

The error analysis of a reduced fourth-order FDS for 1D parabolic problem

Conclusions