This discussion focuses on numerically solving the complex Time Domain Ginzburg-Landau Equation using a Python simulator. The user employs a fourth-order Runge-Kutta (RK4) solver, as the odeint solver does not support complex variables. Key insights include the necessity of separating real and imaginary parts during integration and using discrete convolutional kernels for the Laplacian operator. The conversation also highlights challenges in handling nonlinear terms and the need for validation strategies in simulations.
PREREQUISITESResearchers and developers working on numerical simulations of complex systems, particularly in the fields of superconductivity and fluid dynamics, as well as those interested in advanced numerical methods for PDEs.
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 ).