An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation
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.
scientific article; zbMATH DE number 6940695
scientific article