The discussion revolves around numerically solving the complex Time Domain Ginzburg-Landau Equation (TDGL) using Python. Participants explore methods for implementing a fourth order Runge-Kutta (RK4) algorithm for complex variables, the handling of Laplacians in complex fields, and the challenges of simulating fluxon nucleation in superconductors.
Participants express various viewpoints on the implementation of numerical methods, the handling of complex variables, and the challenges posed by the nonlinear terms in the TDGL equation. There is no consensus on the best approach or method to resolve the issues raised.
Participants highlight limitations in their current methods, such as the handling of the nonlinear term and the potential for overflow in simulations. There are also references to specific mathematical techniques and numerical methods that may not be universally applicable.
Researchers and practitioners interested in numerical methods for solving complex partial differential equations, particularly in the context of superconductivity and related fields.
Twigg said:Finite difference is the method I was thinking of. Specifically, a central-difference scheme for the Laplacian. The catch is that your problem is non-linear due to the |ψ|2ψ|\psi|^{2} \psi term. You will need to use Newton's method to get a solution, and that opens a new can of worms with convergence and validation. I've never tried this personally, but I know at least one piece of commercial software (COMSOL) that uses Newton's method to solve the matrix problems of finite element methods.
You're absolutely right, my mistake. I was thinking of a steady-state problem, which this is not.Strum said:Why would you think there is a problem with non-linear terms? I really can not see how this should pose a problem as long as he uses some explicit time integration scheme ( and even if he used an implicit I can not see why the difficulties would even be related to the finite difference method ).