A nonlinear Galerkin method for numerical approximation of the dynamics of mesophases under flow

Abstract

In recent years, many theoretical and computational studies have evidenced remarkable similarities between the long-time evolution of solutions of dissipative partial differential equations (PDEs) and solutions of finite-dimensional dynamical systems, or ordinary differential equations (ODEs): both possess a finite number of asymptotic degrees of freedom. With the inertial manifold theory, it is shown that certain dissipative PDEs have the same asymptotic behaviour of an appropriate finite dimensional ODEs system. Hence, this theory is extremely promising for numerical studies, because it allows reducing computational effort compared with traditional dynamic reduction procedures. In recent works, new numerical methods, inspired by the inertial manifold theory, such as nonlinear Galerkin methods, are developed. In the present work, we study the dynamics of the so-called Smoluchowski equation for liquid crystalline polymers with the Galerkin-Euler method (one of nonlinear Galerkin methods stemming from the inertial manifold theory), A comparison between the results obtained with traditional procedures and the new technique is developed. Since this method is adequate for the long-time behaviour of dynamical systems, the comparison is performed with bifurcation analysis by using a continuation algorithm.