Aplicación del método local discontinuous galerkin a ecuaciones con derivadas fraccionarias

Abstract

We consider a time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order 1 < α < 2. We obtain optimal rates of convergence for the semi-discrete minimal dissipation Local Discontinuous Galerkin (mdLDG) method by penalizing the primary variable with a term of order h 1−α . Using a von Neumann analy- sis, stability conditions proportional to h α are derived for the forward Euler method and both fractional operators. The CFL condition is numerically studied with respect to the ap- proximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveals that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when α close to one. On the other hand, using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger (FNLS) equation are shown to be conserved for two stabili- zed versions of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank-Nicolson; and the explicit circularly exact Leapfrog sche- me. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann sta- bility analysis.