Modeling and direct numerical simulation of particle-laden flows have a tremendous variety of applications in science and engineering across a vast spectrum of scales from pollution dispersion in the atmosphere, to fluidization in the combustion process, to aerosol deposition in spray medication, along with many others. Due to their strongly nonlinear and multiscale nature, the above complex phenomena still raise a very steep challenge to the most computational methods. In this review, we provide comprehensive coverage of multibody hydrodynamic (MBH) problems focusing on particulate suspensions in complex fluidic systems that have been simulated using hybrid Eulerian-Lagrangian particulate flow models. Among these hybrid models, the Immersed Boundary-Lattice Boltzmann Method (IB-LBM) provides mathematically simple and computationally-efficient algorithms for solid-fluid hydrodynamic interactions in MBH simulations. This paper elaborates on the mathematical framework, applicability, and limitations of various 'simple to complex' representations of close-contact interparticle interactions and collision methods, including short-range interparticle and particle-wall steric interactions, spring and lubrication forces, normal and oblique collisions, and mesoscale molecular models for deformable particle collisions based on hard-sphere and soft-sphere models in MBH models to simulate settling or flow of nonuniform particles of different geometric shapes and sizes in diverse fluidic systems.

In this paper, we present a high-order unified gas-kinetic scheme (UGKS) using the weighted essentially non-oscillatory with adaptive-order (WENO-AO) method for spatial reconstruction and the two-stage fourth-order scheme for time evolution. Since the UGKS updates both the macroscopic flow variables and microscopic distribution function, and provides an adaptive flux function by combining the equilibrium and non-equilibrium parts, it is possible to take separate treatment of the equilibrium and non-equilibrium calculation in the UGKS for the development of high-order scheme. Considering the fact that high-order techniques are commonly applied in the continuum flow simulation with complex structures, and that the rarefied flow structure is usually smooth in the physical space, we apply the high-order techniques in the equilibrium part of the UGKS for the capturing of macroscopic flow evolution, and retain the calculation of distribution function as a second-order method, so that a balance of computational cost and numerical accuracy could be well achieved. The HUGKS has been validated by several numerical test cases, including sine-wave accuracy test, Sod-shock tube, Couette, oscillating Couette, lid-driven cavity and oscillating cavity flow. It is shown that the current method preserves the multiscale property of the original UGKS and obtains accurate solutions in the near continuum regimes.

A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries, and extends the application to a broader range of non-self-adjoint time-dependent differential equations. In addition, PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy. Experiments results on a series of partial differential equations are reported, which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy, convergence speed, and parallel scalability.

In this work, we study gradient-based regularization methods for neural networks. We mainly focus on two regularization methods: the total variation and the Tikhonov regularization. Adding the regularization term to the training loss is equivalent to using neural networks to solve some variational problems, mostly in high dimensions in practical applications. We introduce a general framework to analyze the error between neural network solutions and true solutions to variational problems. The error consists of three parts: the approximation errors of neural networks, the quadrature errors of numerical integration, and the optimization error. We also apply the proposed framework to two-layer networks to derive a priori error estimate when the true solution belongs to the so-called Barron space. Moreover, we conduct some numerical experiments to show that neural networks can solve corresponding variational problems sufficiently well. The networks with gradient-based regularization are much more robust in image applications.

The simulation for particle or soliton propagation based on linear or nonlinear Schrödinger equations on unbounded domains requires the computational domain to be bounded, and therefore, a special boundary treatment such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML) is needed so that the reflections of outgoing waves at the boundary can be minimized in order to prevent the destruction of the simulation. This article presents a new artificial neural network (ANN) method for solving linear and nonlinear Schrödinger equations on unbounded domains. In particular, this method randomly selects training points only from the bounded computational space-time domain, and the loss function involves only the initial condition and the Schrödinger equation itself in the computational domain without any boundary conditions. Moreover, unlike standard ANN methods that calculate gradients using expensive automatic differentiation, this method uses accurate finite-difference approximations for the physical gradients in the Schrödinger equation. In addition, a Metropolis-Hastings algorithm is implemented for preferentially selecting regions of high loss in the computational domain allowing for the use of fewer training points in each batch. As such, the present training method uses fewer training points and less computation time for convergence of the loss function as compared with the standard ANN methods. This new ANN method is illustrated using three examples.

We present a fast iterative solver for scattering problems in 2D, where a penetrable object with compact support is considered. By representing the scattered field as a volume potential in terms of the Green’s function, we arrive at the Lippmann-Schwinger equation in integral form, which is then discretized using an appropriate quadrature technique. The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method (DAFMM). The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel [1], and the construction of low-rank factorizations of the appropriate low-rank matrix sub-blocks is based on our new Nested Cross Approximation (NCA) [2]. The advantage of the NCA described in [2] is that the search space of so-called far-field pivots is smaller than that of the existing NCAs [3, 4]. Another significant contribution of this work is the use of HODLR based direct solver [5] as a preconditioner to further accelerate the iterative solver. In one of our numerical experiments, the iterative solver does not converge without a preconditioner. We show that the HODLR preconditioner is capable of solving problems that the iterative solver can not. Another noteworthy contribution of this article is that we perform a comparative study of the HODLR based fast direct solver, DAFMM based fast iterative solver, and HODLR preconditioned DAFMM based fast iterative solver for the discretized Lippmann-Schwinger problem. To the best of our knowledge, this work is one of the first to provide a systematic study and comparison of these different solvers for various problem sizes and contrast functions. In the spirit of reproducible computational science, the implementation of the algorithms developed in this article is made available at https://github.com/vaishna77/Lippmann_Schwinger_Solver.

This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $α>1.$ The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which exhibits an $O(h^{1/2})$ convergence for all $α ≥ 1.$ It is a little surprising to see that even the lowest-order WG-MFEM $(α = 1)$ experiences a loss of accuracy. This is different from known results for the finite element method (FEM) or the mixed FEM, and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain. Second, we propose a remedy to bring the approximation rate back to optimal by employing two techniques. One is a specially designed boundary correction technique. The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes, which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements. Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all $α.$ Numerical results further confirm this conclusion.

We propose Poisson integrators for the numerical integration of separable Poisson systems. We analyze three situations in which Poisson systems are separated in three ways and Poisson integrators can be constructed by using the splitting method. Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in terms of long-term energy conservation and computational cost. The Poisson integrators are also shown to be more efficient than the canonicalized sympletic methods of the same order.

This paper presents a pseudopotential lattice Boltzmann analysis to show the deficiency of previous pseudopotential models, i.e., inconsistency between equilibrium velocity and mixture velocity. To rectify this problem, there are two strategies: decoupling relaxation time and kinematic viscosity or introducing a system mixture relaxation time. Then, we constructed two modified models: a two-relaxation-time (TRT) scheme and a triple-relaxation-time (TriRT) scheme to decouple the relaxation time and kinematic viscosity. Meanwhile, inspired by the idea of a system mixture relaxation time, we developed three mixture models under different collision schemes, viz. mix-SRT, mix-TRT, and mix-TriRT models. Afterwards, we derived the advection-diffusion equation for the multicomponent system and derived the mutual diffusivity in a binary mixture. Finally, we conducted several numerical simulations to validate the analysis on these models. The numerical results show that these models can obtain smaller spurious currents than previous models and have a wider range for the accessible viscosity ratio with fourth-order isotropy. Compared to previous models, present models avoid complex matrix operations and only fourth-order isotropy is required. The increased simplicity and higher computational efficiency of these models make them easy to apply to engineering and industrial applications.

An extension to rarefied flow regimes of the lattice Boltzmann method-based model for miscible mixtures developed by Vienne et al. (Physical Review E 100.2 (2019): 023309) is presented. The model is applied to study the gas phase separation phenomenon in mixture flows that traditional macroscopic approaches fail to predict. The extension includes a wall function approach with an empirical coefficient to define an effective mean free path in solid geometries, which locally defines the kinematic viscosity and binary diffusion coefficients. The algorithm is also modified by a local multi-relaxation time collision operator and slip boundary conditions at solid walls. Gaseous mixture flow simulations are conducted through a 2D plane microchannel within the slip and early transition flow regimes. Despite a miscible gaseous phase, the mixture loses its homogeneity and independent velocity profiles for each component are observed in the rarefied regime and captured with the current modeling. In addition, the gas separation phenomenon increases with the rarefaction rate and the molecular mass ratio. The individual treatment of the species within mixture flows in the developed lattice Boltzmann model helps understanding the increasing independent behavior of the individual species within the mixture as the regime becomes more rarefied.