Graduate Numerical Solution for Complex PDE (Ginzburg-Landau Eqn.)

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The discussion focuses on numerically solving the complex Time Domain Ginzburg-Landau Equation using a Python simulator to observe fluxon nucleation in a 2D superconductor domain. A fourth-order Runge-Kutta (RK4) solver is being implemented, with participants discussing how to handle complex variables and suggesting trial functions for nonlinear terms. The conversation also addresses the implementation of a complex Laplacian, with advice to separate real and imaginary parts during computation. Concerns about simulation accuracy, memory usage, and the need for validation strategies are raised, alongside suggestions for improving performance and potential GPU utilization. The overall goal is to refine the simulation and validate the results against theoretical expectations.
  • #31
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.

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 ).
 
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  • #32
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 ).
You're absolutely right, my mistake. I was thinking of a steady-state problem, which this is not.
 

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