Towards the solution reconstruction,one of the main steps in Godunov type finite volume scheme,a class of integrated linear reconstruction(ILR)methods has been developed recently,from which the advantages such as para...Towards the solution reconstruction,one of the main steps in Godunov type finite volume scheme,a class of integrated linear reconstruction(ILR)methods has been developed recently,from which the advantages such as parameters free and maximum principle preserving can be observed.It is noted that only time-dependent problems are considered in the previous study on ILR,while the steady state problems play an important role in applications such as optimal design of vehicle shape.In this paper,focusing on the steady Euler equations,we will extend the study of ILR to the steady state problems.The numerical framework to solve the steady Euler equations consists of a Newton iteration for the linearization,and a geometric multigrid solver for the derived linear system.It is found that even for a shock free problem,the convergence of residual towards the machine precision can not be obtained by directly using the ILR.With the lack of the differentiability of reconstructed solution as a partial explanation,a simple Laplacian smoothing procedure is introduced in the method as a postprocessing technique,which dramatically improves the convergence to steady state.To prevent the numerical oscillations around the discontinuity,an efficient WENO reconstruction based on secondary reconstruction is employed.It is shown that the extra two operations for ILR are very efficient.Several numerical examples are presented to show the effectiveness of the proposed scheme for the steady state problems.展开更多
In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial singularity.To derive optima...In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial singularity.To derive optimal order a posteriori error estimates,the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced.By using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived.Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results,with the convergence ofαorder for the nonsmooth case on a uniform mesh.To recover the optimal convergence order 2-αon a nonuniform mesh,we further develop a time adaptive algorithm by means of barrier function recently introduced.The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.展开更多
基金supported by FDCT of the Macao S.A.R.(0082/2020/A2)National Natural Science Foundation of China(Nos.11922120 and 11871489)+4 种基金the Multi-Year Research Grant(No.2020-00265-FST)of University of Macao,and a grant from Department of Science and Technology of Guangdong Province(No.2020B1212030001)The research of Ruo Li was supported by the National Key R&D Program of China,Project Number 2020YFA0712000.The research of Xucheng Meng is partially supported by the National Natural Science Foundation of China(No.12101057)the Scientific Research Fund of Beijing Normal University(No.28704-111032105)the Start-up Research Fund from BNU-HKBU United International College(No.R72021112)。
文摘Towards the solution reconstruction,one of the main steps in Godunov type finite volume scheme,a class of integrated linear reconstruction(ILR)methods has been developed recently,from which the advantages such as parameters free and maximum principle preserving can be observed.It is noted that only time-dependent problems are considered in the previous study on ILR,while the steady state problems play an important role in applications such as optimal design of vehicle shape.In this paper,focusing on the steady Euler equations,we will extend the study of ILR to the steady state problems.The numerical framework to solve the steady Euler equations consists of a Newton iteration for the linearization,and a geometric multigrid solver for the derived linear system.It is found that even for a shock free problem,the convergence of residual towards the machine precision can not be obtained by directly using the ILR.With the lack of the differentiability of reconstructed solution as a partial explanation,a simple Laplacian smoothing procedure is introduced in the method as a postprocessing technique,which dramatically improves the convergence to steady state.To prevent the numerical oscillations around the discontinuity,an efficient WENO reconstruction based on secondary reconstruction is employed.It is shown that the extra two operations for ILR are very efficient.Several numerical examples are presented to show the effectiveness of the proposed scheme for the steady state problems.
基金supported by the Natural Science Foundation of China(Grants 12271367,12071403)by the Shanghai Science and Technology Planning Projects(Grant 20JC1414200).
文摘In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial singularity.To derive optimal order a posteriori error estimates,the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced.By using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived.Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results,with the convergence ofαorder for the nonsmooth case on a uniform mesh.To recover the optimal convergence order 2-αon a nonuniform mesh,we further develop a time adaptive algorithm by means of barrier function recently introduced.The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.
