Finite Element Method for a Space-Fractional Anti-Diffusive Equation

TL;DR

This paper develops a finite element method combined with Crank-Nicolson scheme to numerically solve a nonlinear space-fractional anti-diffusive equation modeling dune morphodynamics, providing stability and error analysis.

Contribution

It introduces a novel numerical scheme for a space-fractional anti-diffusive equation with stability and error estimates, applied to dune morphodynamics.

Findings

The scheme is stable under certain conditions.

Error estimates are derived for the numerical solution.

Numerical examples confirm convergence and accuracy.

Abstract

The numerical solution of a nonlinear and space-fractional anti-diffusive equation used to model dune morphodynamics is considered. Spatial discretization is effected using a finite element method whereas the Crank-Nicolson scheme is used for temporal discretization. The fully discrete scheme is analyzed to determine stability condition and also to obtain error estimates for the approximate solution. Numerical examples are presented to illustrate convergence results.