Nodal Discontinuous Galerkin Methods For Fractional Differential Equations

As a natural extension of classical calculus, fractional calculus has increas-ingly been applied during the last decades. Due to the memory property of fractional operators, the corresponding fractional differential equations are more suitable to describe some process associated with memorability and heredity. The progress that people have made in fractional models inspires us seeking efficient and robust numerical schemes.DGs combine finite volume method and finite element methods of high ef-fectiveness for classical differential equations, owning a lot of advantages, for example, geometry flexibility, mesh adaptation, and parallelization. DGs have two formula:modal representation and nodal representation. The only difference between them lie in basis functions used for approximation. However, both are equivalent mathematically. By now, in literature DGs for fractional differential equations adopts the former formula, thus this paper make some trials in the latter formula for one-and two-dimensional fractional differential equations.Finally, we provide the stability analysis and error estimate of our schemes. In numerical experiments, the convergence rate is optimal in one dimension with the upwind, which confirms theoretical analysis. Due to the uncertain of up-wind’s direction, the results are not expected as α,β are particular large in two dimension. In further discussion, we adopt the central flux, which is more natural and easy to realize, and it shows the optimal convergence rate as well.