An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation (Q1783383)

The paper deals with the numerical solution of the fractional Ginzburg-Landau equation \[ \partial_t u + (\nu + i\eta)(-\Delta)^{\alpha/2} u + (\kappa+ i \zeta) |u|^2 u - \gamma u = 0, \] where \(\alpha\in (1,2]\) is the order of the Riesz fractional derivative. The problem is discretized w.r.t. time by the second-order backward difference formula combined with the explicit second-order Gear's extrapolation. The space discretization is carried out by a fourth-order fractional quasi-compact method. At each time step, only a linear system with a coefficient matrix independent of the time level is solved. The authors derive the convergence rate of order \(O(\tau^2 + h^4)\) without any restriction concerning the size of the mesh and time steps \(h\) and \(\tau\), respectively. Numerical tests are provided to confirm the accuracy and efficiency of the proposed scheme.