A Lagrangian surface hopping algorithm is implemented to study the two dimensional massless Dirac equation for Graphene with an electrostatic potential, in the semiclassical regime. In this problem, the crossing of the energy levels of the system at Dirac points requires a particular treatment in the algorithm in order to describe the quantum transition—characterized by the Landau-Zener probability— between different energy levels. We first derive the Landau-Zener probability for the underlying problem, then incorporate it into the surface hopping algorithm. We also show that different asymptotic models for this problem derived in [O. Morandi, F. Schurrer, J. Phys. A:Math. Theor. 44 (2011) 265301]may give different transition probabilities. We conduct numerical experiments to compare the solutions to the Dirac equation, the surface hopping algorithm, and the asymptotic models of [O. Morandi, F. Schurrer, J. Phys. A: Math. Theor. 44 (2011) 265301].

Multiscale modelling is a powerful technique, which allows for computational efficiency while retaining small-scale details when they are essential for understanding a finer behaviour of the studied system. In the case of materials modelling, one of the effective multiscaling concepts is domain partitioning, which implies the existence of an explicit interface between various material descriptions, for instance atomistic and continuum regions. When dynamic material behaviour is considered, the major problem for this technique is dealing with reflections of high frequency waves from the interface separating two scales. In this article, a new method is suggested, which overcomes this problem for the case of magnetisation dynamics. The introduction of a damping band at the interface between scales, which absorbs high frequency waves, is suggested. The idea is verified using a number of one-dimensional examples with fine/coarse scale discretisation of a continuum problem of spin wave propagation. This work is the first step towards establishing a reliable atomistic/continuum multiscale transition for the description of the evolution of magnetic properties of ferromagnets.

The differential quadrature method has been widely used in scientific and engineering computation. However, for the basic characteristics of time domain differential quadrature method, such as numerical stability and calculation accuracy or order, it is still lack of systematic analysis conclusions. In this paper, according to the principle of differential quadrature method, it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature. Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method, it has been proved that the differential quadrature method is A-stable and s-stage s-order method. On this basis, in order to further improve the accuracy of the time domain differential quadrature method, a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Padé approximations. The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.

We investigate the critical nuclei morphology in phase transformation by combining two effective ingredients, with the first being the phase field modeling of the relevant energetics which has been a popular approach for phase transitions and the second being shrinking dimer dynamics and its variants for computing saddle points and transition states. In particular, the newly formulated generalized shrinking dimer dynamics is proposed by adopting the Cahn-Hilliard dynamics for the generalized gradient system. As illustrations, a couple of typical cases are considered, including a generic system modeling heterogeneous nucleation and a specific material system modeling the precipitate nucleation in FeCr alloys. While the standard shrinking dimer dynamics can be applied to study the non-conserved case of generic heterogeneous nucleation directly, the generalized shrinking dimer dynamics is efficient to compute precipitate nucleation in FeCr alloys due to the conservation of concentration. Numerical simulations are provided to demonstrate both the complex morphology associated with nucleation events and the effectiveness of generalized shrinking dimer dynamics based on phase field models.

In this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

A susceptible–exposed–infectious theoretical model describing Tasmanian devil population and disease dynamics is presented and mathematically analysed using a dynamical systems approach to determine its behaviour under a range of scenarios. The steady states of the system are calculated and their stability analysed. Closed forms for the bifurcation points between these steady states are found using the rate of removal of infected individuals as a bifurcation parameter. A small-amplitude Hopf region, in which the populations oscillate in time, is shown to be present and subjected to numerical analysis. The model is then studied in detail in relation to an unfolding parameter which describes the disease latent period. The model’s behaviour is found to be biologically reasonable for Tasmanian devils and potentially applicable to other species.

The Bay of Bengal is surrounded by coastline except to the south, where there is open sea. The coastline bends most sharply along the coast of Bangladesh, and there are many small and large islands in the off shore region there. In order to incorporate the island boundaries and the curved coastline properly, in any numerical scheme it is often necessary to consider a very fine grid resolution along the coastal belts whereas this is unnecessary away from the coasts. However, a very fine resolution involves more memory and more CPU time in the numerical solution process, and invites numerical instability. On the other hand, boundary-fitted curvilinear grids in hydrodynamic models for coastal seas, bays and estuaries not only fit to the coastline but also render the finite difference schemes simpler and more accurate. In this article, the boundary-fitted curvilinear grids for the model represent the complete boundary of the area considered by four curves defined by four functions, and the four boundaries of two of the larger islands are then represented approximately by two general functions. An appropriate independent coordinate transformation maps the curvilinear physical area to a square domain, and each island boundary is transformed to a rectangle within this square domain. The vertically integrated shallow water equations are transformed to the new space domain, and solved by a regular explicit finite difference scheme. The model is applied to compute the water levels due to astronomical tides, and also the water levels due to surges associated with tropical storms that hit the coast of Bangladesh.

In this paper, a reliable algorithm based on an adaptation of the standard differential transform method is presented, which is the multi-step differential transform method (MSDTM). The solutions of non-linear oscillators were obtained by MSDTM. Figurative comparisons between the MSDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that the proposed technique is a promising tool to solve non-linear oscillators.

Consider n cells, of which some are target cells, and suppose that each cell has a weight. The cells are killed in a sequential manner, with each currently live cell being the next one killed with a probability proportional to its weight. We study the distribution of the number of cells that are alive at the moment when all the target cells have been killed.