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
Studies the different types of multistep discretization of index 3 differential-algebraic equations in Hessenberg form. Existense, uniqueness and influence of perturbations; Local convergence of multistep discretizati...Studies the different types of multistep discretization of index 3 differential-algebraic equations in Hessenberg form. Existense, uniqueness and influence of perturbations; Local convergence of multistep discretization; Details on the numerical tests.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.
文摘Studies the different types of multistep discretization of index 3 differential-algebraic equations in Hessenberg form. Existense, uniqueness and influence of perturbations; Local convergence of multistep discretization; Details on the numerical tests.
