A computational study of superconducting states near the superconducting-normal phase boundary in mesoscopic finite cylinders is presented. The computational approach uses a finite element method to find numerical solutions of the linearized Ginzburg-Landau equation for samples with various sizes, aspect ratios, and cross-sectional shapes, i.e., squares, triangles, circles, pentagons, and four star shapes. The vector potential is determined using a finite element method with two penalty terms to enforce the gauge conditions that the vector potential is solenoidal and its normal component vanishes at the surface(s) of the sample. The eigenvalue problem for the linearized Ginzburg-Landau equations with homogeneous Neumann boundary conditions is solved and used to construct the superconducting-normal phase boundary for each sample. Vortex-antivortex (V-AV) configurations for each sample that accurately reflect the discrete symmetry of each sample boundary were found through the computational approach. These V-AV configurations are realized just within the phase boundary in the magnetic field-temperature phase diagram. Comparisons are made between the results obtained for the different sample shapes.

In this paper we analyze a long standing problem of the appearance of spurious, non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures. The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem. We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coefficients to small k components would lead to the appearance of non-physical solutions. We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution. This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures. Finally, based on the above requirement of small k, we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.

The effect of the equilibrium pair separation on the evolution of cluster structures is investigated based on a new proposed pair potential. The computational results demonstrate that the potential with large equilibrium pair separation stabilizes decahedra and close-packed structures, while disordered structures appear for the potential with small equilibrium pair separation. The icosahedral clusters are dominated in the middle range of equilibrium pair separation. For the small size clusters (N≤24) the dominated structural motif is the polytetrahedra, which is almost independent of the details of the potential.

We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws. The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame, in which the equations are numerically solved. The optimal reference frame moves (locally in time) with the average characteristic speed of the system, and, in this sense, the resulting method is quasi-Lagrangian. This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution. We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method. This leads to a significantly enhanced resolution, especially in the supersonic case. We demonstrate a great potential of the proposed method on a number of numerical examples.

The scope of this paper is to show how a two-scale asymptotic analysis, based on a superposition principle, allows us to derive high order approximate boundary conditions for a scattering problem of a time-harmonic wave by a thin and tangentially periodic multi-layered domain. The periods are assumed of the same order of the thickness. New terms like memory effect and variance-covariance ones are observed contrarily to the laminar case. As a result, optimal error estimates are obtained.

This paper presents an approach to model the solvent-excluded surface (SES) of 3D protein molecular structures using the geometric PDE-based level-set method. The level-set method embeds the shape of 3D molecular objects as an isosurface or level set corresponding to some isovalue of a scattered dense scalar field, which is saved as a discretely-sampled, rectilinear grid, i.e., a volumetric grid. Our level-set model is described as a class of tri-cubic tensor product B-spline implicit surface with control point values that are the signed distance function. The geometric PDE is evolved in the discrete volume. The geometric PDE we use is the mean curvature specified flow, which coincides with the definition of the SES and is geometrically intrinsic. The technique of speeding up is achieved by use of the narrow band strategy incorporated with a good initial approximate construction for the SES. We get a very desirable approximate surface for the SES.

In this paper, a level set method is applied to the inverse problem of 2-D wave equation in the fluid-saturated media. We only consider the situation that the parameter to be recovered takes two different values, which leads to a shape reconstruction problem. A level set function is used to present the discontinuous parameter, and a regularization functional is applied to the level set function for the ill-posed problem. Then the resulting inverse problem with respect to the level set function is solved by using the damped Gauss-Newton method. Numerical experiments show that the method can recover parameter with complicated geometry and the noise in the observation data.

In this paper, a new symmetric energy-conserved splitting FDTD scheme (symmetric EC-S-FDTD) for Maxwell's equations is proposed. The new algorithm inherits the same properties of our previous EC-S-FDTDI and EC-S-FDTDII algorithms: energy-conservation, unconditional stability and computational efficiency. It keeps the same computational complexity as the EC-S-FDTDI scheme and is of second-order accuracy in both time and space as the EC-S-FDTDII scheme. The convergence and error estimate of the symmetric EC-S-FDTD scheme are proved rigorously by the energy method and are confirmed by numerical experiments.

We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost. The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty. Combined with high accuracy of the gPC-based forward solver, the new algorithm can provide great efficiency in practical applications. A rigorous error analysis of the algorithm is conducted, where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate. It is proved that fast (exponential) convergence of the gPC forward solution yields similarly fast (exponential) convergence of the posterior. The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of varying smoothness and dimension.

We present a model for hyperpycnal plumes or turbidity currents that takes into account the interaction between the turbidity current and the bottom, considering deposition and erosion effects as well as solid transport of particles at the bed load due to the current. Water entrainment from the ambient water in which the turbidity current plunges is also considered. Motion of ambient water is neglected and the rigid lid assumption is considered. The model is obtained as a depth-average system of equations under the shallow water hypothesis describing the balance of fluid mass, sediment mass and mean flow. The character of the system is analyzed and numerical simulations are carried out using finite volume schemes and path-conservative Roe schemes.

The parareal algorithm, proposed firstly by Lions et al. [J. L. Lions, Y. Maday, and G. Turinici, A "parareal" in time discretization of PDE's, C.R. Acad. Sci. Paris Sér. I Math., 332 (2001), pp. 661-668], is an effective algorithm to solve the time-dependent problems parallel in time. This algorithm has received much interest from many researchers in the past years. We present in this paper a new variant of the parareal algorithm, which is derived by combining the original parareal algorithm and the Richardson extrapolation, for the numerical solution of the nonlinear ODEs and PDEs. Several nonlinear problems are tested to show the advantage of the new algorithm. The accuracy of the obtained numerical solution is compared with that of its original version (i.e., the parareal algorithm based on the same numerical method).

In this paper we use an analytical-numerical approach to find, in a systematic way, new 1-soliton solutions for a discrete sine-Gordon system in one spatial dimension. Since the spatial domain is unbounded, the numerical scheme employed to generate these soliton solutions is based on the artificial boundary method. A large selection of numerical examples provides much insight into the possible shapes of these new 1-solitons.