Generative adversarial networks (GANs) were initially proposed to generate images by learning from a large number of samples. Recently, GANs have been used to emulate complex physical systems such as turbulent flows. However, a critical question must be answered before GANs can be considered trusted emulators for physical systems: do GANs-generated samples conform to the various physical constraints? These include both deterministic constraints (e.g., conservation laws) and statistical constraints (e.g., energy spectrum of turbulent flows). The latter have been studied in a companion paper (Wu et al., Enforcing statistical constraints in generative adversarial networks for modeling chaotic dynamical systems. Journal of Computational Physics. 406, 109209, 2020). In the present work, we enforce deterministic yet imprecise constraints on GANs by incorporating them into the loss function of the generator. We evaluate the performance of physics-constrained GANs on two representative tasks with geometrical constraints (generating points on circles) and differential constraints (generating divergence-free flow velocity fields), respectively. In both cases, the constrained GANs produced samples that conform to the underlying constraints rather accurately, even though the constraints are only enforced up to a specified interval. More importantly, the imposed constraints significantly accelerate the convergence and improve the robustness in the training, indicating that they serve as a physics-based regularization. These improvements are noteworthy, as the convergence and robustness are two well-known obstacles in the training of GANs.

We introduce novel high order well-balanced finite volume methods for the full compressible Euler system with gravity source term. They require no à priori knowledge of the hydrostatic solution which is to be well-balanced and are not restricted to certain classes of hydrostatic solutions. In one spatial dimension we construct a method that exactly balances a high order discretization of any hydrostatic state. The method is extended to two spatial dimensions using a local high order approximation of a hydrostatic state in each cell. The proposed simple, flexible, and robust methods are not restricted to a specific equation of state. Numerical tests verify that the proposed method improves the capability to accurately resolve small perturbations on hydrostatic states.

Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem.

In calculating the time evolution of an atomic system on diffusive timescales, off-lattice kinetic Monte Carlo (OLKMC) can sometimes be used to overcome the limitations of Molecular Dynamics. OLKMC relies on the harmonic approximation to Transition State Theory, in which the rate of rare transitions from one energy minimum to a neighboring minimum scales exponentially with an energy barrier on the potential energy surface. This requires locating the index-1 saddle point, commonly referred to as a transition state, that separates two neighboring energy minima. In modeling the evolution of an atomic system, it is desirable to find all the relevant transitions surrounding the current minimum. Due to the large number of minima on the potential energy surface, exhaustively searching the landscape for these saddle points is a challenging task. In examining the particular case of isolated Lennard-Jones clusters of around 50 particles, we observe very slow convergence of the total number of saddle points found as a function of successful searches. We seek to understand this behavior by modeling the distribution of successful searches and sampling this distribution to create a stochastic process that mimics this behavior. Finally, we will discuss an improvement to a rejection scheme for OLKMC where we terminate searches that appear to be failing early in the search process.

In this paper, we consider the numerical approximations of a magnetohy-drodynamic potential model that was developed in [15]. Several decoupled, linear, unconditionally energy stable schemes are developed by combining some subtle implicit-explicit treatments for nonlinear coupling terms and the projection-type method for the Navier-Stokes equations. The divergence-free condition for the magnetic field is preserved in the fully-discrete level. We further establish the well-posedness and unconditional energy stabilities of the proposed schemes and present a series of numerical examples in 3D, including accuracy/stability tests, benchmark simulations of driven cavity flow and hydromagnetic Kelvin-Helmholtz instability.

Drag correlations are very important in particle-laden two-phase flow simulations. Some statistical studies have investigated extracting particle configuration factors from simulation data to improve drag correlations. However, little attention has been paid to studying particle configuration effects on drag from the perspective of the flow mechanism. In this paper, a direct numerical simulation (DNS) method based on the second-order accurate immersion interface method is developed to provide highly reliable data. Then, the 'shielding' effect of the two-particle configuration on drag is comprehensively analysed under different angles, distances, and Reynolds number $(Re)$ values, revealing that the complex configuration dependence of the drag influence is attributed to the dominant flow mechanism, such as the 'pressure region unit', 'nozzle', and 'wake' effects. Moreover, we study the 'superposition' effect of the three-particle configuration on drag in a finite $Re$ range. The results show that when the surrounding particles do not directly shield each other, the drag influence calculated by pairwise linear superposition is close to the drag influence revealed by DNS. Otherwise, when the shielding phenomenon of the surrounding particles is obvious and the $Re$ is high, the drag influence of the nearest particle can represent the DNS result.

The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical models that describe the activity of neural networks. These models have been studied at a microscopic level, using Stochastic Differential Equations, and at a mesoscopic/macroscopic level, through the mean field limits using Fokker-Planck type equations. The aim of this paper is to improve their understanding, using a numerical study of their particle systems. This allows us to go beyond the mesoscopic/macroscopic description. We answer one of the most important open questions about these models: what happens after all the neurons in the network fire at the same time? We find that the neural network converges towards its unique steady state, if the system is weakly connected. Otherwise, its behaviour is more complex, tending towards a stationary state or a "plateau" distribution (membrane potentials are uniformly distributed between reset and threshold values). To our knowledge, these distributions have not been described before for these nonlinear models. In addition, we analyse in depth the behaviour of the classical and physical solutions of the Stochastic Differential Equations and, we compare it with what is already known about the classical solutions of Fokker-Planck equation. In this way, our numerical analysis, based on the microscopic scale, allows us to explain not only what happens after the explosion phenomenon, but also, how the physical solutions of the Fokker-Planck equation are. This notion of solution, for the Fokker-Planck equation, has not been studied to date.

In this paper, we develop a novel approach by combining a new robust finite difference Hermite weighted essentially non-oscillatory (HWENO) method [51] with the modified ghost fluid method (MGFM) [25] to simulate the compressible two-medium flow problems. The main idea is that we first use the technique of the MGFM to transform a two-medium flow problem to two single-medium cases by defining the ghost fluids status based on the predicted interface status. Then the efficient and robust HWENO finite difference method is applied for solving the single-medium flow cases. By using immediate neighbor information to deal with both the solution and its derivatives, the fifth order finite difference HWENO scheme adopted in this paper is more compact and has higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu [14]. Furthermore, by combining the HWENO scheme with the MGFM to simulate the two-medium flow problems, less ghost point information is needed than that in using the classical WENO scheme in order to obtain the same numerical accuracy. Various one-dimensional and two-dimensional two-medium flow problems are solved to illustrate the good performances of the proposed method.

In this paper, we consider the field model for complex ionic fluids with an energy variational structure, and analyze the well-posedness to this model with regularized kernels. Furthermore, we deduce the estimate of the maximal density function to quantify the finite size effect. On the numerical side, we adopt a finite volume scheme to the field model, which satisfies the following properties: positivity-preserving, mass conservation and energy dissipation. Besides, series of numerical experiments are provided to demonstrate the properties of the steady state and the finite size effect by showing the equilibrium profiles with different values of the parameter in the kernel.

In this work a simple but accurate shallow model for bedload sediment transport is proposed. The model is based on applying the moment approach to the Shallow Water Exner model, making it possible to recover the vertical structure of the flow. This approach allows us to obtain a better approximation of the fluid velocity close to the bottom, which is the relevant velocity for the sediment transport. A general Shallow Water Exner moment model allowing for polynomial velocity profiles of arbitrary order is obtained. A regularization ensures hyperbolicity and easy computation of the eigenvalues. The system is solved by means of an adapted IFCP scheme proposed here. The improvement of this IFCP type scheme is based on the approximation of the eigenvalue associated to the sediment transport. Numerical tests are presented which deal with large and short time scales. The proposed model allows to obtain the vertical structure of the fluid, which results in a better description on the bedload transport of the sediment layer.

Recently, there is a need for a general-purpose event generator for decays of an elementary particle or a hadron to a state of higher multiplicity $(N ≥ 2)$ that is simple to use and universal. We present the structure of such a library for preparation of generators that generate kinematics of decay processes and can be used to integrate any matrix element squared over phase space of this decay. Some test examples are presented, and results are compared with results known from the literature. As one of examples we consider the Standard Model Higgs boson decay into four leptons. The generators discussed here are compatible with the ROOT interface.