Commun. Comput. Phys., 16 (2014), pp. 1239-1262. New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations Feng Shi 1, Guoping Liang 2, Yubo Zhao 3, Jun Zou 4*1 School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen 518055, China. 2 Beijing FEGEN Software Company, Beijing 100190, China. 3 Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China. 4 Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. Received 3 October 2013; Accepted (in revised version) 3 June 2014 Available online 29 August 2014 doi:10.4208/cicp.031013.030614a 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. AMS subject classifications: 65M12, 65M60, 76D05 Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: Convention-dominated diffusion problems, Navier-Stokes equations, operator splitting, finite elements, multistep scheme. *Corresponding author. Email: shifeng@hitsz.edu.cn (F. Shi), yb.zhao@siat.ac.cn (Y. Zhao), zou@math.cuhk.edu.hk (J. Zou)