Numerical approximation of the Ginzburg–Landau equation with memory effects in the dynamics of phase transitions

We consider the out-of-equilibrium time evolution of a nonconserved order parameter using the Ginzburg–Landau equation including memory effects. Memory effects are expected to play important role on the nonequilibrium dynamics for fast phase transitions, in which the time scales of the rapid phase conversion are comparable to the microscopic time scales. We consider two numerical approximation schemes based on Fourier collocation and finite difference methods and perform a numerical analysis with respect to the existence, stability and convergence of the schemes. We present results of direct numerical simulations for one and three spatial dimensions, and examine numerically the stability and convergence of both approximation schemes.