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We design explicit nonstandard finite difference schemes for the nonlinear Allen–Cahn reaction diffusion equation in the limit of very small interaction length. In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimizing the restrictions normally associated with standard explicit finite difference schemes. The derivation involves splitting the equation into the space-independent and the time-independent different models. An exact nonstandard scheme is proposed for the space-independent model and energy conservative schemes are proposed for the time-independent model. We show the power of the derived scheme over the existing schemes through several numerical examples. [ABSTRACT FROM AUTHOR]

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation. [ABSTRACT FROM AUTHOR]

We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substance as a polymer solution, whose viscoelastic rheology is described by the classical Flory-Huggins theory. We show that one-dimensional solutions exist, which take the form at large times of travelling waves, and we characterize their form and speed in terms of the describing parameters of the problem. Numerical solutions of the time-dependent problem converge to the travelling wave solutions. [ABSTRACT FROM AUTHOR]