Feature identification is an important task in many fluid dynamics applications and diverse methods have been developed for this purpose. These methods are based on a physical understanding of the underlying behavior of the flow in the vicinity of the feature. Particularly, they require the definition of suitable criteria (i.e. point-based or neighborhood-based derived properties) and proper selection of thresholds. However, these methods rely on creative visualization of physical idiosyncrasies of specific features and flow regimes, making them non-universal and requiring significant effort to develop. Here we present a physics-based, data-driven method capable of identifying any flow feature it is trained to. We use convolutional neural networks, a machine learning approach developed for image recognition, and adapt it to the problem of identifying flow features. This provides a general method and removes the large burden placed on identifying new features. The method was tested using mean flow fields from numerical simulations, where the recirculation region and boundary layer were identified in several two-dimensional flows through a convergent-divergent channel, and the horseshoe vortex was identified in three-dimensional flow over a wing-body junction.

A lattice model of interacting fermions is studied with the principal aim of assessing the dependence of calculated mean-field ground states versus the N×N lattice size, with N=16, 32, and 48. A two band model on the two-dimensional square lattice is simulated, with on-site energies and interaction parameters chosen to represent crystal field split orbitals in the moderately correlated regime. Nearest neighbor hopping leads to the well known van Hove singularities (vHs) of the square lattice. Anomalies in the inverse participation ratio of the eigenstates are found to be associated with the vHs, with their prevalence decreasing inversely with N. For the chosen model, inhomogeneous spin densities are always obtained for the small lattice size N=16, with the degree of variation decreasing rapidly for most polarizations as N is increased. Various spin polarizations are treated, and one case in which spin density inhomogeneity persists for the largest lattice size is discussed and analyzed. Coupling of spin density inhomogeneities to charge density variation is minor but evident, and is primarily of intra-orbital origin.

Shallow flow models are used for a large number of applications including weather forecasting, open channel hydraulics and simulation-based natural hazard assessment. In these applications the shallowness of the process motivates depth-averaging. While the shallow flow formulation is advantageous in terms of computational efficiency, it also comes at the price of losing vertical information such as the flow's velocity profile. This gives rise to a model error, which limits the shallow flow model's predictive power and is often not explicitly quantifiable. We propose the use of vertical moments to overcome this problem. The shallow moment approximation preserves information on the vertical flow structure while still making use of the simplifying framework of depth-averaging. In this article, we derive a generic shallow flow moment system of arbitrary order starting from a set of balance laws, which has been reduced by scaling arguments. The derivation is based on a fully vertically resolved reference model with the vertical coordinate mapped onto the unit interval. We specify the shallow flow moment hierarchy for kinematic and Newtonian flow conditions and present 1D numerical results for shallow moment systems up to third order. Finally, we assess their performance with respect to both the standard shallow flow equations as well as with respect to the vertically resolved reference model. Our results show that depending on the parameter regime, e.g. friction and slip, shallow moment approximations significantly reduce the model error in shallow flow regimes and have a lot of potential to increase the predictive power of shallow flow models, while keeping them computationally cost efficient.

In this paper, we consider numerical approximations for the fourth-order Cahn-Hilliard equation with the concentration-dependent mobility and the logarithmic Flory-Huggins bulk potential. One numerical challenge in solving such system is how to develop proper temporal discretization for nonlinear terms in order to preserve its energy stability at the time-discrete level. We overcome it by developing a set of first and second order time marching schemes based on a newly developed "Invariant Energy Quadratization" approach. Its novelty is producing linear schemes, by discretizing all nonlinear terms semi-explicitly. We further rigorously prove all proposed schemes are unconditionally energy stable. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes thereafter.

Based on two-grid discretizations, a local and parallel finite element algorithm (LPFEA) based on Newton iteration for solving the stationary incompressible magnetohydrodynamics (MHD) is considered in this paper. The basic idea of the algorithm is to compute the nonlinear system by Newton iteration on a globally coarse mesh first, then solve a series of subproblems of residual correction on the corresponding subdomains with fine grids in parallel. The optimal error estimates with respective to iterative step m and mesh sizes H and h≪H are derived. The efficiency of the method is illustrated by numerical experiments.

In this paper, we first discuss the existence and uniqueness of a class of nonlinear saddle-point problems, which are frequently encountered in physical models. Then, a generalized Arrow-Hurwicz method is introduced to solve such problems. For the method, the convergence rate analysis is established under some reasonable conditions. It is also applied to solve three typical discrete methods in fluid computation, with the computational efficiency demonstrated by a series of numerical experiments.

The exact Riemann solver for one-dimensional elastic-perfectly plastic solid has been presented in the previous work [S. Gao and T. G. Liu, Adv. Appl. Math. Mech., 9(3), 2017, 621-650], but its iterative process of finding nonlinear equation solution is time-consuming. In this paper, to enhance the computational efficiency of the exact Riemann solver and provide a more practical Riemann solver for actual implementation, we design a non-iterative solution structure-based adaptive approximate (SSAA) Riemann solver for one-dimensional elastic-perfectly plastic solid. Judging the solution structure adaptively and then solving the Riemann problem with corresponding solution structure non-iteratively can shorten the computing time and meanwhile guarantee the correctness of the final result. Numerical performance tests manifest that the exact Riemann solver is indeed time-consuming and the ordinary approximate Riemann solver with fixed three-wave solution structure is of great error, whereas the SSAA Riemann solver is of both efficiency and accuracy. Error estimation further indicates that the SSAA Riemann solver has at least second-order accuracy to approach the exact solution of the states in the star region.

We describe the algorithms used in the Matlab continuation and bifurcation package pde2path for Hopf bifurcation and continuation of branches of periodic orbits in systems of PDEs in 1, 2, and 3 spatial dimensions, including the computation of Floquet multipliers. We first test the methods on three reaction diffusion examples, namely a complex Ginzburg-Landau equation as a toy problem, a reaction diffusion system on a disk with rotational waves including stable spirals bifurcating out of the trivial solution, and a Brusselator system with interaction of Turing and Turing-Hopf bifurcations. Then we consider a system from distributed optimal control, which is ill-posed as an initial value problem and thus needs a particularly stable method for computing Floquet multipliers, for which we use a periodic Schur decomposition. The implementation details how to use pde2path on these problems are given in an accompanying tutorial, which also includes a number of further examples and algorithms, for instance on Hopf bifurcation with symmetries, on Hopf point continuation, and on branch switching from periodic orbits.

In this paper we propose an extremum-preserving iterative procedure for the imperfect interface problem. This method is based on domain decomposition method. First we divide the domain into two sub-domains by the interface, then we alternately solve the sub-domain problems with Robin boundary condition. We prove that the iterative method is convergent and the iterative procedure is extremum-preserving at PDE level. At last, some numerical tests are carried out to demonstrate the convergence of the iterative method by using a special discrete method introduced on sub-domains.

To implement the Wigner branching random walk, the particle carrying a signed weight, either −1 or +1, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from −1 to +1. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adjust the skewness of particle weights. Performance evaluations on the Gaussian barrier scattering (2D) and a Helium-like system (4D) demonstrate the feasibility of both strategies and the variance reduction property of the second approach. We provide an improvement of the first signed-particle implementation that partially alleviates the restriction on the time step and perform a thorough theoretical and numerical comparison among all the existing signed-particle implementations. Details on implementing the importance sampling according to the quasi-probability density and an efficient resampling or particle reduction are also provided.

In this paper, we describe our initiative to utilize a modern well-tested numerical platform in the field of material physics: the Genetic Hybrid Algorithm (GHA). Our aim is to develop a powerful special-purpose tool for finding ground state structures. Our task is to find the diamond bulk atomic structure of a silicon supercell through optimization. We are using the semi-empirical Tersoff potential. We focus on a 2x2x1 supercell of cubic silicon unit cells; of the 32 atoms present, we have fixed 12 atoms at their correct positions, leaving 20 atoms for optimization. We have been able to find the known global minimum of the system in different 19-, 43- and 60-parameter cases. We compare the results obtained with our algorithm to traditional methods of steepest descent, simulated annealing and basin hopping. The difficulties of the optimization task arise from the local minimum dense energy landscape of materials and a large amount of parameters. We need to navigate our way efficiently through these minima without being stuck in some unfavorable area of the parameter space. We employ different techniques and optimization algorithms to do this.

We develop a numerical scheme for nonlinear Poisson-Boltzmann equation. First, we regularize the solution of PBE to remove the singularity. We introduce the discontinuous bubble function to treat the nonhomogeneous jump conditions of the regularized solution. Next, starting with an initial guess, we apply linearization to treat the nonlinearity. Then, we discretize the discontinuous bubble and the bilinear form of PBE. Finally, we solve the discretized linear problem by IFEM. This process is repeated by updating the previous approximation. 

We carry out numerical experiments. We observe optimal convergence rate for all examples.