Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation

•An efficient solver for the time-fractional heat equation discretized on non- uniform temporal meshes is proposed.•The solver is a space-time multigrid method based on the waveform relaxation technique.•Hierarchical matrix approximations are considered to maintain optimal complexity of the algorith...

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Bibliographic Details
Published in:Journal of computational physics Vol. 416; p. 109540
Main Authors: Hu, Xiaozhe, Rodrigo, Carmen, Gaspar, Francisco J.
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 01.09.2020
Elsevier Science Ltd
Subjects:
Algorithms
Computational physics
Differential equations
Discretization
Graded meshes
Hierarchical matrices
Multigrid waveform relaxation
Operators (mathematics)
Semi-algebraic mode analysis
Thermodynamics
Time-fractional heat equation
Waveforms
Multigrid waveform relaxation
Hierarchical matrices
Graded meshes
Time-fractional heat equation
Semi-algebraic mode analysis
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:•An efficient solver for the time-fractional heat equation discretized on non- uniform temporal meshes is proposed.•The solver is a space-time multigrid method based on the waveform relaxation technique.•Hierarchical matrix approximations are considered to maintain optimal complexity of the algorithm.•A semi-algebraic analysis supports the numerical findings. This work deals with the efficient numerical solution of the time-fractional heat equation discretized on non-uniform temporal meshes. Non-uniform grids are essential to capture the singularities of “typical” solutions of time-fractional problems. We propose an efficient space-time multigrid method based on the waveform relaxation technique, which accounts for the nonlocal character of the fractional differential operator. To maintain an optimal complexity, which can be obtained for the case of uniform grids, we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix (H-matrix) representation. In particular, the proposed method has a computational cost of O(kNMlog⁡(M)), where M is the number of time steps, N is the number of spatial grid points, and k is a parameter which controls the accuracy of the H-matrix approximation. The efficiency and the good convergence of the algorithm, which can be theoretically justified by a semi-algebraic mode analysis, are demonstrated through numerical experiments in both one- and two-dimensional spaces.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2020.109540