In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,th...In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,the stability and convergence analysis are well established in[Liao and Zhang,Math.Comp.,90(2021),1207–1226]and[Chen,Yu,and Zhang,arXiv:2108.02910,2021].However,the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels.By developing a novel spectral norm inequality,the unconditional stability and convergence are rigorously proved under the updated step ratio restriction rk:=τk/τk−1≤1.405 for BDF3 method.Finally,numerical experiments are performed to illustrate the theoretical results.To the best of our knowledge,this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.展开更多
In this paper we propose a continuous downscaling data assimilation algorithm for solving reaction-diffusion equations with a critical parameter.For the spatial discretization we consider the finite element methods.Tw...In this paper we propose a continuous downscaling data assimilation algorithm for solving reaction-diffusion equations with a critical parameter.For the spatial discretization we consider the finite element methods.Two backward differentiation formulae(BDF),a backward Euler method and a two-step backward differentiation formula,are employed for the time discretization.Employing the dissipativity property of the underlying reaction-diffusion equation,under suitable conditions on the relaxation(nudging)parameter and the critical parameter,we obtain uniform-in-time error estimates for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements given on a coarse mesh by an interpolation operator.Numerical experiments verify and complement our theoretical results.展开更多
基金supported by the Science Fund for Distinguished Young Scholars of Gansu Province(Grant No.23JRRA1020)the Fundamental Research Funds for the Central Universities(Grant No.lzujbky-2023-06).
文摘In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,the stability and convergence analysis are well established in[Liao and Zhang,Math.Comp.,90(2021),1207–1226]and[Chen,Yu,and Zhang,arXiv:2108.02910,2021].However,the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels.By developing a novel spectral norm inequality,the unconditional stability and convergence are rigorously proved under the updated step ratio restriction rk:=τk/τk−1≤1.405 for BDF3 method.Finally,numerical experiments are performed to illustrate the theoretical results.To the best of our knowledge,this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.
基金supported by grants from the National Natural Science Foundation of China(Grant No.12271367)Shanghai Science and Technology Planning Projects(Grant No.20JC1414200).
文摘In this paper we propose a continuous downscaling data assimilation algorithm for solving reaction-diffusion equations with a critical parameter.For the spatial discretization we consider the finite element methods.Two backward differentiation formulae(BDF),a backward Euler method and a two-step backward differentiation formula,are employed for the time discretization.Employing the dissipativity property of the underlying reaction-diffusion equation,under suitable conditions on the relaxation(nudging)parameter and the critical parameter,we obtain uniform-in-time error estimates for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements given on a coarse mesh by an interpolation operator.Numerical experiments verify and complement our theoretical results.
