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Related Experiment Video

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All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations.

Marco Donatelli1, Rolf Krause2, Mariarosa Mazza1

  • 1University of Insubria, via Valleggio 11, 22100 Como, Italy.

Calcolo
|November 22, 2021
PubMed
Summary
This summary is machine-generated.

We developed efficient parallel solvers for space-fractional diffusion equations. Our methods leverage a two-level Toeplitz structure for computational cost independent of time, showing optimal performance with specific time discretization schemes.

Keywords:
All-at-once systemsFractional diffusion equationsSpace–time multigridSpectral distributionToeplitz matrices

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Fractional diffusion equations model anomalous diffusion processes.
  • Efficient numerical methods are crucial for solving these complex equations.
  • Discretization often leads to large, structured linear systems.

Purpose of the Study:

  • To develop parallelizable, time-independent computational cost solvers for time-dependent space-fractional diffusion equations.
  • To investigate the impact of multigrid strategies on solving these systems.
  • To analyze the influence of time discretization schemes on solver performance.

Main Methods:

  • All-at-once discretization of the time-dependent fractional diffusion equation.
  • Exploiting the two-level Toeplitz structure of the resulting linear system.
  • Applying and analyzing multigrid methods with semi- and full-coarsening.
  • Evaluating different time discretization schemes, including Crank-Nicolson and second-order backward-difference.

Main Results:

  • The two-level Toeplitz structure enables the construction of efficient iterative solvers.
  • Multigrid methods show sensitivity to the chosen time discretization.
  • The Crank-Nicolson scheme hinders multigrid convergence, while the second-order backward-difference scheme offers good convergence under specific conditions.
  • The proposed methods are effective for both constant and variable diffusion coefficients, and in 2D.

Conclusions:

  • Efficient, parallel solvers for space-fractional diffusion problems can be developed by exploiting matrix structures.
  • The choice of time discretization is critical for the performance of multigrid solvers.
  • The second-order backward-difference scheme is recommended for its stability and convergence properties in this context.