In this paper, we focus on one difficulty arising in the numerical simulation of the Vlasov-Poisson system: when using a regular grid-based solver with periodic boundary conditions, perturbations present at the initial time artificially reappear at a later time. For regular finite-element mesh in velocity, we show that this recurrence time is actually linked to the spectral accuracy of the velocity quadrature when computing the charge density. In particular, choosing trigonometric quadrature weights optimally defers the occurrence of the recurrence phenomenon. Numerical results using the Semi-Lagrangian Discontinuous Galerkin and the Finite Element/Semi-Lagrangian method confirm the analysis.

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L² error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$)  norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.

It is found that imaginary time propagation method can effectively deliver a convergent result in solving Kohn–Sham equation, but a sufficient long simulation is needed to reach an accurate enough result, while the self-consistent field iteration method for Kohn–Sham equation can be more efficient when it works, but it sometimes suffers from divergence. In this work, we take advantage of the convergence of imaginary time propagation method by generating a quality initial guess to improve the behavior of self-consistent field iteration. A number of numerical experiments successfully show that i). for those self-consistent field iterations which are sensitive to the initial guess, the results obtained from imaginary time propagation method make the iterations converge, and ii). generally, the convergence of self-consistent field iteration can be accelerated by imaginary time propagation method. It is shown that all-electron models can be resolved well with the proposed method.

Molecular statics (MS) based on energy minimization serves as a useful simulation technique to study mechanical behaviors and structures at atomic level. The efficiency of MS, however, still remains a challenge due to the complexity of mathematical optimization in large dimensions. In this paper, the Inertia Accelerated Molecular Statics (IAMS) method is proposed to improve computational efficiency in MS simulations. The core idea of IAMS is to let atoms move to meta positions very close to their final equilibrium positions before minimization starts at a specific loading step. It is done by self-learning from historical movements (atomic inertia effect) without knowledge of external loadings. Examples with various configurations and loading conditions indicate that IAMS can effectively improve efficiency without loss of fidelity. In the simulation of three-point bending of nanopillar, IAMS shows efficiency improvement of up to 23 times in comparison with original MS. Particularly, the size-independent efficiency improvement makes IAMS more attractive for large-scale simulations. As a simple yet efficient method, IAMS also sheds light on improving the efficiency of other energy minimization-based methods.

Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, we show that these models lack global hyperbolicity and the loss of hyperbolicity already occurs for small deviations from equilibrium. This leads to instabilities in a numerical test case. We then derive new Hyperbolic Shallow Water Moment Equations based on a modification of the system matrix. The model can be written in analytical form and hyperbolicity can be proven for a large number of equations. A second variant of this model is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem using the new models yield accurate and fast solutions while guaranteeing hyperbolicity.

In the present paper we analyze and discuss some mathematical aspects of the fluid-static configurations of a self-gravitating perfect gas enclosed in a spherical solid shell. The mathematical model we consider is based on the well-known Lane-Emden equation, albeit under boundary conditions that differ from those usually assumed in the astrophysical literature. The existence of multiple solutions requires particular attention in devising appropriate numerical schemes apt to deal with and catch the solution multiplicity as efficiently and accurately as possible. In sequence, we describe some analytical properties of the model, the two algorithms used to obtain numerical solutions, and the numerical results for two selected cases.

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. In this paper, we construct a lowest-order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. The scheme is based on Lagrange finite element and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the non-self-adjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed problem which does not fit into the framework of classical theoretical analysis. Instead, we obtain the convergence analysis based on the spectral approximation theory of compact operator. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.

A new method is proposed to simulate a coupled air-water two-layer turbulent channel flow. A numerically effective dynamic viscosity is implemented to calculate the viscous momentum flux at the interface, leading to a strong-coupling scheme for the evolution of air and water motions. The direct numerical simulation results are compared with those in the literature obtained from a weak-coupling scheme. It is discovered that while the turbulence statistics of the air phase based on the strong- and weak-coupling schemes are close to each other, the results on the water side are influenced by the coupling approach, especially near the water surface.

We consider a model of coupled free and porous media flow governed by Stokes equation and Darcy's law with the Beavers-Joseph-Saffman interface condition. In this paper, we propose a new numerical approach for the Stokes-Darcy system. The approach employs the classical finite element method for the Darcy region and the weak Galerkin finite element method for the Stokes region. We construct corresponding discrete scheme and prove its well-posedness. The estimates for the corresponding numerical approximation are derived. Finally, we present some numerical experiments to validate the efficiency of the approach for solving this problem.

Flux coordinates and field-line coordinates are ubiquitous in magnetically confined plasma research. Most of these coordinates are essentially non-orthogonal curvilinear coordinates, in which the differential operators are rather complicated. This article reports an automatic tool, OpGen, for generating a finite difference coefficient matrix for field solvers by using a computer symbolic computation system. This tool is suitable for, but is not limited to, code development for toroidally confined plasmas.

In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D multi-group neutron transport equation with delayed neutron is developed. The explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order Crank-Nicholson scheme, become the subsets of the current IUGKS. In neutron transport, the microscopic angular flux and the macroscopic scalar flux are fully coupled in an implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step. Mathematically, the current scheme has the asymptotic preserving (AP) property in recovering automatically the diffusion solution in the continuum regime. Since the explicit scanning along neutron traveling direction within the computational domain is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other schemes, such as the explicit UGKS, the commonly used finite difference, and finite volume methods. This study shows that the IUGKS can be used faithfully to study neutron transport in practical engineering applications.

The vast diversity in neuron cell morphology has led to an increase in automated algorithms which can accurately reconstruct neurons from microscopy images. The poor quality of brightfield and fluorescence microscopy images and the thin branch-like fibrous structure of neurons make the process of manual segmentation challenging. We propose a novel automatic neuron segmentation framework using a B-spline based active contour deformation model with hyperelastic regularization, and develop a MATLAB software tool named "NeuronSeg_BACH". In NeuronSeg_BACH, initialization of the contour is done automatically by detecting cell body and neurites separately. This boundary-extraction based algorithm utilizes cubic B-splines to deform active contours to match the neuron cell surface accurately. Using adaptive local refinement, finer level deformation of the active contour is captured using truncated hierarchical B-splines in a multiresolution manner. By introducing hyperelastic regularization, we allow large nonlinear deformations of the active contours. Unlike other existing methods which represent boundaries as piecewise constant functions, we provide a more accurate and smooth representation of the neuron geometry. In the level set segmentation framework, the implicit level set function is defined using $C^2$ continuous B-splines. Improved segmentation results are shown for 2D and 3D synthetic and microscopy images as compared to other methods.