Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection

In this work, we approximate a time-dependent problem with drift involving fractional powers of elliptic operators. The numerical scheme is based on an integral representation of the stationary problem at each time step. The integral representation is further approximated by an exponentially convergent sinc quadrature. This results in multiple independent reaction-diffusion problems approximated using the finite element method. The resulting error between the solution and its approximation in the energy norm is based on a Strang’s lemma for the consistency errors generated by sinc quadrature and finite element approximations. The L 2 error is obtained by a standard duality argument. A forward Euler method is considered for the time stepping. It is stable provided the sinc quadrature stepping and time stepping is taken sufficiently small. Under the same condition, we also deduce its first order convergence in time. We challenge the analyzed numerical scheme in the context of surface quasi-geostrophic (SQG) dynamics and electroconvection system. In each setting, the governing equations are derived from conservations of various physical quantities. In SQG dynamics the drifting velocity involves the solution to another fractional elliptic problem. The simulations consider two scenarios: inviscid (no diffusion) and inviscid-limit (small diffusion). In both scenarios our simulation results are compared with existing results and good agreements are observed. In electroconvection, the liquid is located in between two concentric circular electrodes which are either assumed to be of infinite height or slim. Each configuration results in a different nonlocal electro-magnetic model defined on a two dimensional bounded domain. Our numerical simulations indicate that slim electrodes are favorable for electroconvection to occur and are able to sustain the phenomena over long period of time. Furthermore, we provide a numerical study on the influence of the three main parameters of the system: the Rayleigh number, the Prandtl number and the electrodes aspect ratio