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Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equation with nonsmooth data [article]

Binjie Li, Tao Wang, Xiaoping Xie
2019 arXiv   pre-print
This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems.  ...  Nearly optimal convergence rate with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate ln(1/τ)(√(ln(1/h)) h^2+τ) is derived  ...  Conclusion A time-stepping discontinuous Galerkin method is analyzed in this paper.  ... 
arXiv:1908.09189v1 fatcat:qo5f2xlponbwbnpweikxfvkkuq

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  ...  Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order α∈(0,1) in time, due to their many successful applications in  ...  methods for fractional evolution equations.  ... 
doi:10.1016/j.cma.2018.12.011 fatcat:gpcmkn5quzeztkw6s6qbaltzwy

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.  ...  A detailed comparison with several popular time stepping schemes is also performed.  ...  The authors are grateful to the anonymous referees for their constructive comments. The research of B.  ... 
arXiv:1404.3800v4 fatcat:dtrrca3slzcbvigrzcofb55qpi

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  
In this paper we focus on the fractional cases 0 < α < 1 and 1 < α < 2, with a Caputo derivative, which are known as the subdiffusion and diffusion-wave equation, respectively, in the literature.  ...  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 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.  ...  We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods  ...  [33] to construct a second order time discretisation scheme for solving diffusion wave equation, see (3.30) and (3.31) in Section 3 below.  ... 
doi:10.1515/fca-2017-0058 fatcat:cncu4n73vfcvxlgffhreeuel5a

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.  ...  Let us give a brief introduction of two kinds of numerical methods for solving fractional diffusion equations with nonsmooth data: the L1-type method [14, 19, 9, 32, 18] , discontinuous Galerkin method  ... 
arXiv:1908.09145v2 fatcat:qziyxcgitramjadr3vey5tyone

Optimal control of flow with discontinuities

Chris Homescu, I.M. Navon
2003 Journal of Computational Physics  
Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model.  ...  Optimal control of the 1-D Riemann problem of Euler equations is studied, with the initial values for pressure and density as control parameters.  ...  Wesseling for sharing their expertise. The authors thank their reviewers for their insightful comments, which proved to be very useful for a better presentation of the results.  ... 
doi:10.1016/s0021-9991(03)00154-2 fatcat:txsqe4xjvzdhlb6u23o4zvwqlu

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

S.T. Miller, R.B. Haber
2008 Computer Methods in Applied Mechanics and Engineering  
Yang [22] uses a characteristicsbased TVD scheme for one-dimensional problems. In [23], the method is extended to two dimensions using a fractional step method.  ...  Eqs. (1) and (2) can either be solved as a system or, given second-order differentiability for both T and q and spatial uniformity of s, be combined into a single wave equation with a damping term: In  ...  Support was provided by the Center for Process Simulation and Design (CPSD) and the Materials Computation Center (MCC) at the University of Illinois at Urbana-Champaign under NSF Grant numbers ITR/AP DMR  ... 
doi:10.1016/j.cma.2008.07.016 fatcat:kgl6f7bmpvdwjnt3zszjj7ncam

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Chi-Wang Shu
2016 Journal of Computational Physics  
In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous  ...  For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite  ...  Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Mathematics of Computation, 77 (2008), 699-730. [65] M.  ... 
doi:10.1016/j.jcp.2016.04.030 fatcat:ucrmechmcfcitdjhvb3ynd2g6e

Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data [article]

Bangti Jin, Raytcho Lazarov, Dongwoo Sheen, Zhi Zhou
2015 arXiv   pre-print
In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation.  ...  We develop a space semidiscrete scheme based on the standard Galerkin finite element method, and establish error estimates optimal with respect to data regularity in L^2(D) and H^1(D) norms for both smooth  ...  Acknowledgments The research of R. Lazarov and Z. Zhou have been supported in parts by NSF Grant DMS-1016525 while that of D. Sheen by NRF-2014R1A2A1A11052429.  ... 
arXiv:1504.01529v1 fatcat:jusuq2zonvdz3calureiiuhiv4

Meshless techniques for anisotropic diffusion

Annamaria Mazzia, Giorgio Pini, Flavio Sartoretto
2014 Applied Mathematics and Computation  
A good numerical method would be locally mass conservative, produce no or minimal over/under-shoots, produce minimal numerical diffusion, and require no CFL time-step limit for stability.  ...  The latter would allow better use of parallel computers, since time-stepping is essentially a serial process. Moreover, it would be good for the methods to be of high order accuracy.  ...  We show how to solve the equations using a global implicit approach in an efficient way, and we present the derived computational results.  ... 
doi:10.1016/j.amc.2014.03.032 fatcat:c527226gyfgbffnq4p67qxd7wi

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

Pin Lyu, Seakweng Vong
2021 arXiv   pre-print
We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations.  ...  An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations.  ...  Lately, for the problem with nonsmooth data, a Petrov-Galerkin method and a time-stepping discontinuous Galerkin method are proposed in [22] (Luo, Li and Xie) and [14] (Li, Wang and Xie), where  ... 
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

Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems [article]

Samir Karaa
2020 arXiv   pre-print
We derive optimal L^2-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order α∈ (0,1), for cases with smooth and nonsmooth initial data.  ...  A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods.  ...  For other types of time-fractional problems, one may refer to [5] for fractional diffusion-wave equations and to [28] for an integro-differential equation.  ... 
arXiv:2004.12113v1 fatcat:m6hqjz3zlbdxzcri77onvaddke

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  ...  We shall provide a complete error analysis of the scheme under realistic regularity assumptions by means of a novel energy argument.  ...  For the fractional order α → 1, the factor recovers that for the classical diffusion equation [17] .  ... 
arXiv:1508.06134v2 fatcat:adaaztfrpngjpi3dl7qlk3buwa
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