Efficient numerical schemes for two-dimensional Ginzburg-Landau equation in superconductivity

Abstract

The objective of this paper is to propose some high-order compact schemes for two-dimensional Ginzburg-Landau equation. The space is approximated by high-order compact methods to improve the computational efficiency. Based on Crank-Nicolson method in time, several temporal approximations are used starting from different viewpoints. The numerical characters of the new schemes, such as the existence and uniqueness, stability, convergence are investigated. Some numerical illustrations are reported to confirm the advantages of the new schemes by comparing with other existing works. In the numerical experiments, the role of some parameters in the model is considered and tested.

Complex Ginzburg–Landau Equation with Generalized Finite Differences

“…For instance, in [1], the authors used the Radial Basis Functions method, for t = 2 s, and the errors are in accordance with our numerical examples. In addition, in [22], the authors use several compact finite difference schemes and their results are similar to ours, as it is clear from Table 3. The following tables (Tables 4 and 5) collect the numerical convergence order for the three previous clouds and times 0.25, 0.5 and 2 s, computed as error i−1 error i .…”

“…With election of the domain and the clouds of nodes we make clear the potential of the method stated in the introduction. We present a comparison between the results obtained by using the GFDM in this paper and the ones obtained in [1,22]. The first cloud, with 55 nodes, is obtained by distributing the points randomly and deleting the ones which are sufficiently near.…”