GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Sparse matrices in matlab: design and implementation
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
Numerical solution of the space fractional Fokker-Planck equation
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
Finite difference approximations for fractional advection-dispersion flow equations
Journal of Computational and Applied Mathematics
SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
Finite difference methods for two-dimensional fractional dispersion equation
Journal of Computational Physics
A second-order accurate numerical method for the two-dimensional fractional diffusion equation
Journal of Computational Physics
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
Finite difference approximations for two-sided space-fractional partial differential equations
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
A Space-Time Spectral Method for the Time Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
New Trends in Nanotechnology and Fractional Calculus Applications
New Trends in Nanotechnology and Fractional Calculus Applications
A direct O(Nlog2N) finite difference method for fractional diffusion equations
Journal of Computational Physics
An implicit RBF meshless approach for time fractional diffusion equations
Computational Mechanics
Fractional Calculus for Scientists and Engineers
Fractional Calculus for Scientists and Engineers
Multigrid method for fractional diffusion equations
Journal of Computational Physics
Novel Numerical Methods for Solving the Time-Space Fractional Diffusion Equation in Two Dimensions
SIAM Journal on Scientific Computing
Fractional Dynamics and Control
Fractional Dynamics and Control
Journal of Computational and Applied Mathematics
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The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.