Finite difference method

Abstract: A numerical method to simulate discharging processes in mass‐flow silos is presented. The essential point is to formulate the appropriate constitutive law for a granular bulk material, which covers solid‐like as well as fluid‐like behavior during discharging. An elastic‐plastic law is chosen for the former one, which is completed with a simple first approach for fluid‐like behavior. As large and fast deformations occur, geometric nonlinearities and mass properties of the bulk material are considered with respect to an Eulerian frame of reference. The complete set of field equations is numerically solved by the finite element method spatially and by the finite difference method in time. Due to the nature of the finite element method a broad variety of boundary conditions can be studied. The method provides transient velocity and stress fields within the bulk material for a first period of discharging. Remarkable stress redistributions with strong increases of wall pressures are computed.

Abstract: We adapt a finite difference method of solution of the two-dimensional massless Dirac equation, developed in the context of lattice gauge theory, to the calculation of electrical conduction in a graphene sheet or on the surface of a topological insulator The discretized Dirac equation retains a single Dirac point (no ``fermion doubling''), avoids intervalley scattering as well as trigonal warping, and preserves the single-valley time-reversal symmetry $(=\text{symplectic}\text{ }\text{symmetry})$ at all length scales and energies\char22{}at the expense of a nonlocal finite difference approximation of the differential operator We demonstrate the symplectic symmetry by calculating the scaling of the conductivity with sample size, obtaining the logarithmic increase due to antilocalization We also calculate the sample-to-sample conductance fluctuations as well as the shot-noise power and compare with analytical predictions

Abstract: A newly disclosed nonstandard finite difference method has been used to discretize a Lotka–Volterra model to investigate the critical normal form coefficients of bifurcations for both one-parameter and two-parameter bifurcations. The discrete-time prey–predator model exhibits a variety of local bifurcations such as period-doubling, Neimark–Sacker, and strong resonances. Critical normal form coefficients are determined to reveal dynamical scenarios corresponding to each bifurcation point. We also investigate the complex dynamics of the model numerically by Matlab package using MatcotM based on numerical continuation technique. The numerical continuation validates the theoretical analysis, which is discussed from an ecological perspective.