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New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

Published online by Cambridge University Press:  03 June 2015

Feng Shi ,
Guoping Liang ,
Yubo Zhao  and
Jun Zou
Feng Shi*
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen 518055, China
Guoping Liang
Affiliation:
Beijing FEGEN Software Company, Beijing 100190, China
Yubo Zhao*
Affiliation:
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China
Jun Zou*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
*
Email:shifeng@hitsz.edu.cn
Email:yb.zhao@siat.ac.cn
Corresponding author.Email:zou@math.cuhk.edu.hk
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Abstract

We present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.

Keywords

65M1265M6076D05Convention-dominated diffusion problemsNavier-Stokes equationsoperator splittingfinite elementsmultistep scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 5 , November 2014 , pp. 1239 - 1262
Copyright
Copyright © Global Science Press Limited 2014

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