The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial...The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial singularity.In the numerical simulations of solutions with initial singularity,variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence,whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied.In this paper,we revisit the variable time-step implicit-explicit two-step backward differentiation formula(IMEX BDF2)scheme to solve the parabolic integro-differential equations(PIDEs)with initial singularity.We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_(k):=T_(k)/T_(k-1)<r_(max)=4.8645(k≥3)and a much mild requirement on the first ratio,i.e.r_(2)>0.This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy,i.e.the graded mesh t_(k)=T(k/N)^(γ).In this situation,the convergence order of O(N^(-min(2,γα))is achieved,where N denotes the total number of mesh points andαindicates the regularity of the exact solution.This is,the optimal convergence will be achieved by taking%γ_(opt)=2/α.Numerical examples are provided to demonstrate our theoretical analysis.展开更多
基金supported by the NSFC(Grant No.12171376)the Fundamental Research Funds for the Central Universities(Grant No.2042021kf0050)and WHU-2022-SYJS-0002+1 种基金Yana Di is partially supported by the NSFC(Grant Nos.12271048,12171042)the Guangdong Key Laboratory(Grant No.2022B1212010006).
文摘The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial singularity.In the numerical simulations of solutions with initial singularity,variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence,whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied.In this paper,we revisit the variable time-step implicit-explicit two-step backward differentiation formula(IMEX BDF2)scheme to solve the parabolic integro-differential equations(PIDEs)with initial singularity.We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_(k):=T_(k)/T_(k-1)<r_(max)=4.8645(k≥3)and a much mild requirement on the first ratio,i.e.r_(2)>0.This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy,i.e.the graded mesh t_(k)=T(k/N)^(γ).In this situation,the convergence order of O(N^(-min(2,γα))is achieved,where N denotes the total number of mesh points andαindicates the regularity of the exact solution.This is,the optimal convergence will be achieved by taking%γ_(opt)=2/α.Numerical examples are provided to demonstrate our theoretical analysis.
