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An efficient solving method for nonlinear convection diffusion equation

Xia Cui (Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China)
GuangWei Yuan (Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China)
ZhiJun Shen (Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 2 January 2018

149

Abstract

Purpose

This paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated.

Design/methodology/approach

A nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy.

Findings

Theoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained.

Practical implications

The numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems.

Originality/value

The FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

Keywords

Acknowledgements

The authors are grateful to Professor Longjun Shen for his helpful suggestions. This work is supported by the National Natural Science Foundation of China (11271054, 11571048, 11471048, U1630249), the Science Foundation of CAEP (2014A0202010), Yu Min Foundation, the Science Challenge Project (JCKY2016212A502) and the Foundation of LCP.

Citation

Cui, X., Yuan, G. and Shen, Z. (2018), "An efficient solving method for nonlinear convection diffusion equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 28 No. 1, pp. 173-187. https://doi.org/10.1108/HFF-10-2016-0403

Publisher

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Emerald Publishing Limited

Copyright © 2018, Emerald Publishing Limited

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