Linear Implicit Variable-step BDF Method For Incompressible Navier-Stokes Equations

Navier-Stokes equations are equations of motion that describe the conservation of momentum in viscous incompressible fluids.Because of their wide application in practical problems,they have important practical significance for the study of numerical solutions of the equations.In this paper,linearly implicit backward differential formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations(formulated in vorticity-stream function).With linearly implicit backward differential formulas with variable step-sizes for time discretization,the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations.As a result,only linear solvers are needed at each time step to achieve the desired temporal accuracy.Due to the FFT's convenience,the Fourier pseudospectral methods are used in spatial discretization.With the help of a priori assumption and aliasing error control techniques,the error estimates for one-step and two-step backward differential formulas are established in several norms under appropriate step-size constraints.This paper also carried out the numerical experiments under the strong viscosity coefficient and the weak viscosity coefficient respectively: Compared with the constant step-sizes BDF methods,the methods in this paper has smaller errors and the best convergence order;compared with the implicit-explicit BDF2 methods(explicit processing of nonlinear convection terms),the methods has better stability;compared with the fully implicit Crank-Nicolson methods,the computational cost of the methods in this paper are smaller.This proves the effectiveness and robustness of the methods proposed in this paper.Because the a posterior error estimate is based on the adaptive algorithm,this paper further proposes quadratic reconstructions based on the linear implicit variable step-sizes BDF methods,and obtains the corresponding a posterior error estimate.The experimental results show the correctness and consistency of the theoretical analysis.