Finite Element Method for a Space-Fractional Anti-Diffusive Equation
This work provides a numerical framework for a specific class of fractional PDEs in geophysical modeling, but the method is an incremental adaptation of existing techniques to a new equation.
The authors developed and analyzed a finite element method combined with Crank-Nicolson time stepping for a nonlinear space-fractional anti-diffusive equation modeling dune morphodynamics, providing stability conditions and error estimates with numerical convergence results.
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.