Sweeping algorithms for inverting the discrete Ginzburg-Landau operator

摘要

The Ginzburg Landau equations we study arise in the modeling of superconductivity. One widely used method of discretizing the equations together with the associated periodic boundary conditions, in the case of a rectangle of dimension 2, leads to a five-point stencil. Solving the system means inverting a sparse matrix of dimension N2, where N is the number of grid points on each side of the rectangle. We propose a method that is similar to the shooting technique in the numerical solution of ordinary differential equations. For small N, the method requires inverting a full matrix of dimension 2N. When N is large, an iterative procedure combining partial sweeping and the technique of divide and conquer (domain decomposition) is appropriate.