Numerical Modelling of Domain Structure Annihilation in a Nonlinear Hyperbolic Phase-Field Model

Abstract

Context and relevance. Modelling the dynamics of the order parameter in nonlinear media with inertial effects is of interest for condensed matter physics and the theory of phase transitions. Hyperbolic generalisations of the Landau–Khalatnikov equation make it possible to account for wave mechanisms of energy transport, which fundamentally distinguishes them from classical relaxation models. Their numerical investigation requires a stable and consistent second-order accurate algorithm capable of correctly reproducing both nonlinear relaxation and inertially induced wave processes. Objective. To investigate the dynamics of the order parameter in a nonlinear medium described by an inertial equation of the Landau–Khalatnikov–Tani type, with application to the process of spontaneous annihilation of a narrow-localised domain structure in a two-dimensional formulation. Hypothesis. It is assumed that inclusion of the inertial term leads to a qualitative modification of the domain boundary evolution, namely to redistribution of gradient energy into dynamical oscillations of the order parameter field, and that the use of an implicit finite-difference scheme with iterative treatment of the nonlinearity ensures correct reproduction of all stages of annihilation. Methods and materials. An implicit Crank–Nicolson finite-difference scheme with second-order temporal accuracy is employed for the numerical solution. The nonlinear term is approximated using the Newton method with iterative refinement at each time step. Correctness of the algorithm implementation is confirmed by comparison of numerical and analytical solutions for a test problem with a known exact solution. Results. Numerical experiments demonstrate successive stages of domain structure evolution: convergence of the interfaces, their disappearance, and the formation of propagating wave disturbances. It is shown that the presence of the inertial term leads to redistribution of part of the gradient energy into dynamical oscillations of the order parameter field. Conclusions. The obtained results confirm the correctness of the implemented numerical approach and demonstrate the significant influence of inertial effects on the dynamics of localised structures. The hyperbolic formulation allows description of wave processes absent in parabolic models, which is important for analysing the energy balance and the evolution of domain boundaries.