A fast method for solving time-dependent nonlinear convection diffusion problems

Qian He; Wenxin Du; Feng Shi; Jiaping Yu; Qian He; Wenxin Du; Feng Shi; Jiaping Yu

doi:10.3934/era.2022109

Electronic Research Archive

2022, Volume 30, Issue 6: 2165-2182. doi: 10.3934/era.2022109

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A fast method for solving time-dependent nonlinear convection diffusion problems

1.
College of Science, Harbin Institute of Technology, Shenzhen 518055, China
2.
College of Science, Donghua University, Shanghai 201620, China

Academic Editor: Jingzhi Li

Received: 31 December 2021 Revised: 07 April 2022 Accepted: 12 April 2022 Published: 18 April 2022

In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.
- nonlinear convection diffusion problems,
- operator splitting,
- multistep technique,
- finite element method,
- Burgers type equation,
- Buckley-Leverett type equation
Citation: Qian He, Wenxin Du, Feng Shi, Jiaping Yu. A fast method for solving time-dependent nonlinear convection diffusion problems[J]. Electronic Research Archive, 2022, 30(6): 2165-2182. doi: 10.3934/era.2022109

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Abstract

In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.

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