An efficient solving method for nonlinear convection diffusion equation

作者：Cui Xia^*; Yuan GuangWei; Shen ZhiJun

来源：International Journal of Numerical Methods for Heat and Fluid Flow, 2018, 28(1): 173-187.

DOI：10.1108/HFF-10-2016-0403

• Convergence
• Numerical analysis
• Convection diffusion equation
• High efficiency
• Iteration acceleration
• Nonlinear scheme

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 L1 (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.

• 出版日期2018
• 单位北京应用物理与计算数学研究所