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

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.