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Analysis of the L1 scheme for fractional wave equations with nonsmooth data [article]

Binjie Li, Tao Wang, Xiaoping Xie
2019 arXiv   pre-print
This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data.  ...  In addition, a modified L1 scheme is proposed, and stability and temporal accuracy O(τ^2) are derived for this scheme with nonsmooth initial data.  ...  As far as we know, the convergence in the L 2 (Ω)-norm of the L1 scheme for fractional wave equations with nonsmooth data has not been established.  ... 
arXiv:1908.09145v2 fatcat:qziyxcgitramjadr3vey5tyone

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Neville J. Ford, Yubin Yan
2017 Fractional Calculus and Applied Analysis  
AbstractIn this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data.  ...  The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy.  ...  We next consider the corrected L1 scheme (3.26)- (3.28) . By Theorem 3.2, the convergence rate of the corrected L1 scheme (3.26)-(3.28) is O(τ 2−α ) for smooth and nonsmooth data.  ... 
doi:10.1515/fca-2017-0058 fatcat:cncu4n73vfcvxlgffhreeuel5a

Two Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data [article]

Bangti Jin, Raytcho Lazarov, Zhi Zhou
2015 arXiv   pre-print
We consider the initial/boundary value problem for the fractional diffusion and diffusion-wave equations involving a Caputo fractional derivative in time.  ...  The numerical results indicate that the proposed fully discrete schemes are accurate and robust for nonsmooth data, and competitive with existing schemes.  ...  The authors are grateful to the anonymous referees for their constructive comments. The research of B.  ... 
arXiv:1404.3800v4 fatcat:dtrrca3slzcbvigrzcofb55qpi

Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

Bangti Jin, Raytcho Lazarov, Zhi Zhou
2019 Computer Methods in Applied Mechanics and Engineering  
The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal  ...  We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes  ...  However, the rigorous convergence analysis of such schemes can be very challenging, and is mostly missing for nonsmooth problem data. 5.2. Petrov-Galerkin formulation.  ... 
doi:10.1016/j.cma.2018.12.011 fatcat:gpcmkn5quzeztkw6s6qbaltzwy

Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

Bangti Jin, Raytcho Lazarov, Zhi Zhou
2016 SIAM Journal on Scientific Computing  
Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.  ...  These two schemes are first and second-order accurate in time for both smooth and nonsmooth data.  ...  The authors are grateful to the anonymous referees for their constructive comments. The research of B.  ... 
doi:10.1137/140979563 fatcat:odijwwnzavbtvcykhth25hp6qq

An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data [article]

Bangti Jin and Raytcho Lazarov and Zhi Zhou
2015 arXiv   pre-print
In this work, we revisit the error analysis of the scheme, and establish an O(τ) convergence rate for both smooth and nonsmooth initial data.  ...  The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time.  ...  Acknowledgment The authors are grateful to the anonymous referee for his/her constructive comments, which have led to improved quality of the paper. The research of B.  ... 
arXiv:1501.00253v1 fatcat:xcoyyl4qhrcgpaist5z67hp4bq

Error estimates for a robust finite element method of two-term time-fractional diffusion-wave equation with nonsmooth data

Lijuan Nong, An Chen, Jianxiong Cao
2021 Mathematical Modelling of Natural Phenomena  
Error estimates for semidiscrete and fully discrete schemes are established with respect to nonsmooth data.  ...  In this paper, we consider a two-term time-fractional diffusion-wave equation which involves the fractional orders $\alpha\in (1, 2)$ and $\beta\in(0, 1)$, respectively.  ...  The first two authors' work was funded by the Guangxi Natural Science Foundation under grant numbers 2018GXNSFBA281020 and 2018GXNSFAA138121, and the Doctoral Starting up Foundation of Guilin University  ... 
doi:10.1051/mmnp/2021007 fatcat:nvpeb5kqcvenlnhkwri4moo6am

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

Na An
2020 Advances in Difference Equations  
Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional  ...  In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time t = 0 $t=0$ .  ...  Availability of data and materials Not applicable.  ... 
doi:10.1186/s13662-020-02976-4 doaj:ffd8c9c1f56548728e2345ff59616ce5 fatcat:pjhnch5wufgg3gynssrmorhog4

A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations [article]

Pin Lyu, Seakweng Vong
2021 arXiv   pre-print
An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations.  ...  We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations.  ...  analysis of the associated fast schemes.  ... 
arXiv:2101.09678v3 fatcat:eyidssjtejcs3a3hew2sbarthy

An analysis of galerkin proper orthogonal decomposition for subdiffusion

Bangti Jin, Zhi Zhou
2016 Mathematical Modelling and Numerical Analysis  
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order α ∈ (0, 1) in time, which is often used to describe anomalous diffusion processes  ...  We shall provide a complete error analysis of the scheme under realistic regularity assumptions by means of a novel energy argument.  ...  Acknowledgements The work of the first author (B. Jin) is partly supported by EPSRC grant EP/M025160/1. The authors are grateful to the referees for their constructive comments.  ... 
doi:10.1051/m2an/2016017 fatcat:xhplcu223jgylgqjkcouph6b5y

Complete monotonicity-preserving numerical methods for time fractional ODEs [article]

Lei Li, Dongling Wang
2020 arXiv   pre-print
., 76(1):189-198, 2018), we show that the ℒ1 scheme is 𝒞ℳ-preserving, so that the ℒ1 scheme is at least A(π/2) stable, which is an improvement on stability analysis for ℒ1 scheme given in Jin, Lazarov  ...  The results for fractional ODEs are extended to 𝒞ℳ-preserving numerical methods for Volterra integral equations with general completely monotone kernels.  ...  An analysis of the modified l1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM Journal on Numerical Analysis, 56(1):210–227, 2018. [Zbi11] C. J.  ... 
arXiv:1909.13060v3 fatcat:wdwetrbezvhuvgvamvbiwgdgoa

An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion [article]

Bangti Jin, Zhi Zhou
2016 arXiv   pre-print
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order α∈ (0,1) in time, which is often used to describe anomalous diffusion processes  ...  Extensive numerical experiments are presented to verify the convergence analysis and the efficiency of the proposed scheme.  ...  The convergence rate of the L1 scheme improves with the smoothness of the initial data v (while keeping the smooth right hand side f fixed) and the increase of the fractional order α, since the solution  ... 
arXiv:1508.06134v2 fatcat:adaaztfrpngjpi3dl7qlk3buwa

Finite Volume Element Methods for Two-Dimensional Time Fractional Reaction-Diffusion Equations on Triangular Grids [article]

Zhichao Fang
2021 arXiv   pre-print
In this paper, the time fractional reaction-diffusion equations with the Caputo fractional derivative are solved by using the classical L1-formula and the finite volume element (FVE) methods on triangular  ...  The existence and uniqueness for the fully discrete FVE scheme are given.  ...  [9] revisited the error analysis of the L1-formula, especially for the nonsmooth initial data, the authors obtained O(τ ) convergence rate. Li et al.  ... 
arXiv:2101.12541v1 fatcat:lxxijgauongohku3qhojvliho4

Numerical analysis of two new finite difference methods for time-fractional telegraph equation

Xiaozhong Yang, ,School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China, Xinlong Liu
2017 Discrete and continuous dynamical systems. Series B  
Under the premise of smooth solution, theoretical analysis and numerical experiments show that the E-I and I-E difference schemes are unconditionally stable, with 2nd order spatial accuracy, 2 − α order  ...  Fractional telegraph equations are an important class of evolution equations and have widely applications in signal analysis such as transmission and propagation of electrical signals.  ...  The authors would like to express their sincere thanks to the editor and anonymous referees for insightful comments and suggestions which have led to improvements in presentation of this manuscript.  ... 
doi:10.3934/dcdsb.2020269 fatcat:iaibusvt5rhthdbxc5dtq3tcvy

Correction of high-order BDF convolution quadrature for fractional evolution equations [article]

Bangti Jin, Buyang Li, Zhi Zhou
2017 arXiv   pre-print
The desired k^ th-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth.  ...  Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.  ...  Acknowledgements The authors are grateful to Professor Christian Lubich for his valuable comments on an earlier version of the paper. The work of B. Jin is supported by UK EPSRC grant EP/M025160/1.  ... 
arXiv:1703.08808v1 fatcat:2j5qvwfg6rawtg6ot7aey6cqaa
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