L1-Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by L1-type formula and space discretized by the multiscale Galerkin method. The optimal...

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Bibliographic Details
Published in:Journal of function spaces 2021, Vol.2021
Main Authors: Chen, Jian, Huang, Yong, Zeng, Taishan
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Augmentation
Boundary conditions
Boundary value problems
Burgers equation
Computer applications
Convergence
Discretization
Galerkin method
Inequality
Laplace transforms
Methods
Nonlinear equations
Numerical analysis
Polynomials
Theoretical analysis
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Summary:In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by L1-type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach Oτ2−α+hr in the L2 norm and Oτ2−α+hr−1 in the H1 norm, respectively, in which τ is the time step size, h is the space step size, and r is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.
ISSN:2314-8896
2314-8888
DOI:10.1155/2021/5581102