Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations

A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algor...

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Vydáno v:Journal of scientific computing Ročník 80; číslo 1; s. 1 - 25
Hlavní autoři: Liao, Hong-lin, Yan, Yonggui, Zhang, Jiwei
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.07.2019
Springer Nature B.V
Témata:
Algorithms
Approximation
Boundary value problems
Computational efficiency
Computational Mathematics and Numerical Analysis
Energy methods
Error analysis
Finite element method
Formulas (mathematics)
Linearization
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Theoretical
Semilinear subdiffusion equation
energy method
Discrete
Discrete fractional Grönwall inequality
Two-level L1 formula
Unconditional convergence
ISSN:0885-7474, 1573-7691
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Abstract A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost O ( M N 2 ) and storage O ( MN ) for the standard L1 formula to O ( M N log N ) and O ( M log N ) , respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time t = 0 , and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete H 2 energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.
AbstractList A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost O(MN2) and storage O(MN) for the standard L1 formula to O(MNlogN) and O(MlogN), respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time t=0, and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete H2 energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.
A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost O ( M N 2 ) and storage O ( MN ) for the standard L1 formula to O ( M N log N ) and O ( M log N ) , respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time t = 0 , and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete H 2 energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.
Author Liao, Hong-lin
Yan, Yonggui
Zhang, Jiwei
Author_xml – sequence: 1
  givenname: Hong-lin
  orcidid: 0000-0003-0777-6832
  surname: Liao
  fullname: Liao, Hong-lin
  organization: Department of Mathematics, Nanjing University of Aeronautics and Astronautics
– sequence: 2
  givenname: Yonggui
  surname: Yan
  fullname: Yan, Yonggui
  organization: Beijing Computational Science Research Center (CSRC)
– sequence: 3
  givenname: Jiwei
  orcidid: 0000-0002-4493-7431
  surname: Zhang
  fullname: Zhang, Jiwei
  email: jiweizhang@whu.edu.cn
  organization: School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University
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Keywords Semilinear subdiffusion equation
energy method
Discrete
Discrete fractional Grönwall inequality
Two-level L1 formula
Unconditional convergence
Language English
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Snippet A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion...
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SubjectTerms Algorithms
Approximation
Boundary value problems
Computational efficiency
Computational Mathematics and Numerical Analysis
Energy methods
Error analysis
Finite element method
Formulas (mathematics)
Linearization
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Theoretical
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Title Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations
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