arXiv subjects

Joaquín Quintana-Murillo

Publications and source records attributed to Joaquín Quintana-Murillo.

2 recordsLinked to original sources

arXiv2024

An Adaptive Difference Method for Variable-Order Diffusion Equations

An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of these timesteps is chosen by an adaptive algorithm in order to keep the local error bounded around a preset value, a value that can be chosen at will. For some types of problems this adaptive method is much faster than the corresponding usual method with fixed timesteps while keeping the local error of the numerical solution around the preset values. These findings turns out to be similar to those found for constant-order fractional diffusion equations.

math.NA↗

arXiv2013

A Finite Difference Method with Non-uniform Timesteps for Fractional Diffusion Equations

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behaviour of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.

math.NA↗