An overview of new tools in Nektar++ for solving fractional differential equations

Max Carlson (University of Utah, USA)

There exist natural phenomena such as anomalous diffusion that have apparent non-local behavior that can be more faithfully represented using fractional-order differential equations. Since these fractional operators are non-local, traditional discretizations for ODEs and PDEs will not be sparse and require super-linear computational complexity to solve. Because of this property, fast algorithms are crucial for computing accurate solutions to fractional differential equations in a reasonable amount of time. We present two new fast algorithms developed for Nektar++ that can be used to solve differential equations that are fractional-order in time and in space. The first is the FractionalInTime integration scheme that enables time-stepping for ODEs and PDEs with a fractional-order time derivative. The second is a discretization of the spectral fractional Laplacian which is a spatial operator that can be used, among other things, for modelling certain types of anomalous diffusion. The time-fractional integration scheme is currently live on the main Nektar++ branch while the fractional Laplacian work is still experimental and will hopefully be added to the main Nektar++ branch at a future date.