Finite-element method in the space-time domain for nonstationary electrodynamics

A new numerical method for solving problems of nonstationary electrodynamics is suggested that uses integro-differential equations and finite-element representation of fields. Examples of numerical calculations are given.     

An algorithm for LES of premixed compressible flows using the Conditional Moment Closure model

A novel numerical method has been developed to couple a recent high order accurate fully compressible upwind method with the Conditional Moment Closure combustion model. The governing equations, turbulence modelling and numerical methods are presented in full. The new numerical method is validated against direct numerical simulation (DNS) data for a lean premixed methane slot burner. Although the modelling approaches are based on non-premixed flames and hence not expected to be valid for a wide range of premixed flames, the predicted flame is just 10% longer than that in the DNS and excellent agreement of mean mass fractions, conditional mass fractions and temperature is demonstrated. This new numerical method provides a very useful framework for future application of CMC to premixed as well as non-premixed combustion.     

A numerical scheme for the Green–Naghdi model

In this paper a hybrid numerical method using a Godunov type scheme is proposed to solve the Green–Naghdi model describing dispersive “shallow water” waves. The corresponding equations are rewritten in terms of new variables adapted for numerical studies. In particular, the numerical scheme preserves the dynamics of solitary waves. Some numerical results are shown and compared to exact and/or experimental ones in different and significant configurations. A dam-break problem and an impact problem where a liquid cylinder is falling to a rigid wall are solved numerically. This last configuration is also compared with experiments leading to a good qualitative agreement.     

A lattice Bhatnagar--Gross--Krook model for a class of the generalized Burgers equations

A new lattice Bhatnagar--Gross--Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.     

Two New Simple Multi-Symplectic Schemes for the Nonlinear Schrodinger Equation

We investigate the multi-symplectic Euler-box scheme for the nonlinear Schroedinger equation. Two new simple semi-explicit scheme are derived. A composition scheme based on the new derived schemes is also discussed. Some numerical results are reported to illustrate the efficiency of the new schemes.     

Novel Computation of the Growth Rate of Generalized Random Fibonacci Sequences

A new formulation is proposed for the computation of average growth rates of generalized random Fibonacci sequences. Based on the new formula, a novel numerical scheme is designed and successfully implemented, and interesting analytic asymptotic expansions are obtained for several examples.     

A conservative numerical scheme for solving an autonomous Hamiltonian system

A new numerical scheme is proposed for solving Hamilton’s equations that possesses the properties of symplecticity. Just as in all symplectic schemes known to date, in this scheme the conservation laws of momentum and angular momentum are satisfied exactly. A property that distinguishes this scheme from known schemes is proved: in the new scheme, the energy conservation law is satisfied for a system of linear oscillators. The new numerical scheme is implicit and has the third order of accuracy with respect to the integration step. An algorithm is presented by which the accuracy of the scheme can be increased up to the fifth and higher orders. Exact and numerical solutions to the two-body problem, calculated by known schemes and by the scheme proposed here, are compared.     

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

A semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.     

Reducible two-photon exchange diagrams and reference state contribution to energy corrections within the modified adiabatic approach

A new adiabatic method of calculation of QED corrections to the energy of bound states is applied to the derivation of the working formulae for the reducible (reference-state correction) two-photon exchange diagrams. States of two-electron multicharged ions with unequal one-electron energies are considered. The final Feynman gauge expressions are written in a new simple form, convenient for numerical applications.     

Multisymplectic Euler Box Scheme for the KdV Equation

We investigate the multisymplectic Euler box scheme for the Korteweg-de Vries （KdV） equation. A new completely explicit six-point scheme is derived. Numerical experiments of the new scheme with comparisons to the Zabusky-Kruskal scheme, the multisymplectic 12-point scheme, the narrow box scheme and the spectral method are made to show nice numerical stability and ability to preserve the integral invariant for long-time integration.     

Half-Moment Closure for Radiative Transfer Equations

In this paper a moment method for radiative transfer equations is considered which has been developed and investigated using different approaches. Problems appearing for this moment system for boundary value problems using Maxwell-type boundary conditions are described. A new method based on the consideration of positive and negative half fluxes is developed and shown to overcome the above problems. Moreover, a numerical scheme and numerical results for the new moment system are presented.     

New approach to the normal mode method in underwater acoustics

A new approach to the numerical solution of normal model problems in underwater acoustics is presented,in which the corresponding normal mode problem is transformed to the problem of solving a dynamic system.Three applications are considered:(1)the broad band normal mode problem;(2) the range-dependent problem with perturbation proportional to the range parameter;and (3) the evolution of the normal mode with environmental parameters.A numerical simulation for a broad band problem is performed,and the calculated eigenvalues have good agreement with those obtained by the standard normal mode code KRAKAN.     

Variational multiscale element free Galerkin method for the water wave problems

A new numerical method, which is based on the coupling between variational multiscale method and meshfree methods, is developed for the water wave problems, in which the free surface capturing technique is used to capture the position of the free surface. The proposed method takes full advantage of meshfree methods, therefore, no mesh generation and mesh reconstruction are involved. Meanwhile, due to that the proposed method belongs to meshfree methods, thus it is suitable for the highly deformed free surface flow problems. Finally, two water wave problems are solved and the results have also been analyzed. The numerical results show that the proposed method can indeed obtain accurate numerical results for the water wave problems, which does not refer to the choice of a proper stabilization parameter.     

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

A quantum mechanical expression for the lattice contribution to the total dielectric function of semiconductors

Effects of electron-electron and electron-phonon optical interactions on the lattice dielectric function of the doped polar semiconductors are investigated. A new expression for the lattice contribution to the dielectric function is derived using the remarkable Zubarev double-time Green function. A numerical calculation for the case of GaN is done to highlight the accuracy of the model. The results obtained are in agreement with the available experimental data and reproduce the main features observed in Raman scattering spectra.     

On the beam dynamics optimization problem for an alternating-phase focusing linac

A new algorithm for generation a synchronous phase sequence in alternating-phase focusing (APF) linacs is proposed. The optimization of the intense deuteron beam is considered. The results of the numerical optimization are presented.     

A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions.     

On the computation of sound in free and wall-bounded domains

A new Dispersion-Relation-Preserving (DRP) scheme has been developed using the Lax-Wendroff methodology. Two collocated grids are placed in a staggered formation and a staggered DRP scheme is used to calculate the spatial differentiation of the propagation and convection terms. A staggered filtering scheme of a six points stencil is developed to complete the transformation from one grid to another. Existing DRP Runge-Kutta schemes are used for the time marching. Stability limits and accuracy issues are investigated using a simple 1D advection equation. The new method is then tested for monopole and quadrupole radiation, diffraction effects of an aperture in a wall, and convection effects of shear flow. All demonstrate the good accuracy and numerical stability of the new method.     

A new control method based on the lattice hydrodynamic model considering the double flux difference

A new feedback control method is derived based on the lattice hydrodynamic model in a single lane. A signal based on the double flux difference is designed in the lattice hydrodynamic model to suppress the traffic jam. The stability of the model is analyzed by using the new control method. The advantage of the new model with and without the effect of double flux difference is explored by the numerical simulation. The numerical simulations demonstrate that the traffic jam can be alleviated by the control signal.     

Wasserstein distances in the analysis of time series and dynamical systems

A new approach based on Wasserstein distances, which are numerical costs of an optimal transportation problem, allows us to analyze nonlinear phenomena in a robust manner. The long-term behavior is reconstructed from time series, resulting in a probability distribution over phase space. Each pair of probability distributions is then assigned a numerical distance that quantifies the differences in their dynamical properties. From the totality of all these distances a low-dimensional representation in a Euclidean space is derived, in which the time series can be classified and statistically analyzed. This representation shows the functional relationships between the dynamical systems under study. It allows us to assess synchronization properties and also offers a new way of numerical bifurcation analysis.The statistical techniques for this distance-based analysis of dynamical systems are presented, filling a gap in the literature, and their application is discussed in a few examples of datasets arising in physiology and neuroscience, and in the well-known Hénon system.