Mathematical and numerical modelling of non-Brownian diffusion

Transport dynamics in complex systems is often observed to deviate from the standard laws. For instance, in the field of environmental and geophysical fluid dynamics, several studies have highlighted non-Brownian transport dynamics. Discrepancies can occur both for the time relaxation that can deviate from the classical exponential Debye pattern and for the spatial diffusion that can deviate from Fick’s second law. The resulting transport process exhibits a non-linear growth in time of the mean square displacement. If the growth rate is faster than linear, the transport process is superdiffusive and if it is slower than linear, the transport process is sub-diffusive. On the one hand, superdiffusion is characterised by a heavy-tailed jump length distributions resulting in diverging spatial moments and a non-Gaussian dy- namics. On the other hand, subdiffusion is characterised by a heavy-tailed waiting time distribution resulting in diverging temporal moments and a non-Markovian dynamics. Anomalous diffusion and non-exponential relaxation patterns can be described by a space-time fractional-order diffusion equation. Unlike integer-order derivatives that are local operators, fractional-order derivatives are non-local, integro-differential operators. As such, they can be used to represent memory effects and long-range dispersion processes. Space-time fractional-order diffusion models have received an increasing attention in recent years and have been used to model a wide range of problems in surface and subsurface hydrology, plasma turbulence, finance, ecology and epidemiology. In this talk, we will discuss numerical methods to solve the space-time frac- tional diffusion equation. We show that numerical methods like the finite difference and finite element methods are not well suited to solve this equation as they are generally of low order and thus require a lot of grid points to obtain an accurate solution. For local differential operators, this results in large sparse matrices that cannot be handled easily. However, for global differential operators, like the ones considered here, the resulting matrix is full as the global behavior of the function has to be taken into account. High-order, global numerical methods like the pseudo-spectral (PS) method therefore appear to be a better choice as they naturally take the global behavior of the solution into account and use a limited number of degrees of freedom. We will consider flexible numerical discretizations that combine a PS expansion of Mittag-Leffler functions in time with either finite-difference, finite-element or PS discretizations in space