A projection-based reduced order model (ROM) based on the Fourier collocation method is proposed for compressible flows. The incorporation of localized artificial viscosity model and filtering is pursued to enhance the robustness and accuracy of the ROM for shock-dominated flows. Furthermore, for Euler systems, ROMs built on the conservative and the skew-symmetric forms of the governing equation are compared. To ensure efficiency, the discrete empirical interpolation method (DEIM) is employed. An alternative reduction approach, exploring the sparsity of viscosity is also investigated for the viscous terms. A number of one- and two-dimensional benchmark cases are considered to test the performance of the proposed models. Results show that stable computations for shock-dominated cases can be achieved with ROMs built on both the conservative and the skew-symmetric forms without additional stabilization components other than the viscosity model and filtering. Under the same parameters, the skew-symmetric form shows better robustness and accuracy than its conservative counterpart, while the conservative form is superior in terms of efficiency.

Aggregation equations are broadly used to model population dynamics with nonlocal interactions, characterized by a potential in the equation. This paper considers the inverse problem of identifying the potential from a single noisy spatial-temporal process. The identification is challenging in the presence of noise due to the instability of numerical differentiation. We propose a robust model-based technique to identify the potential by minimizing a regularized data fidelity term, and regularization is taken as the total variation and the squared Laplacian. A split Bregman method is used to solve the regularized optimization problem. Our method is robust to noise by utilizing a Successively Denoised Differentiation technique. We consider additional constraints such as compact support and symmetry constraints to enhance the performance further. We also apply this method to identify time-varying potentials and identify the interaction kernel in an agent-based system. Various numerical examples in one and two dimensions are included to verify the effectiveness and robustness of the proposed method.

We present a novel efficient implementation of the flexible boundary condition (FBC) method, initially proposed by Sinclair et al., for large single-periodic problems. Efficiency is primarily achieved by constructing a hierarchical matrix ($\mathscr{H}$-matrix) representation of the periodic Green matrix, reducing the complexity for updating the boundary conditions of the atomistic problem from quadratic to almost linear in the number of pad atoms. In addition, our implementation is supported by various other tools from numerical analysis, such as a residual-based transformation of the boundary conditions to accelerate the convergence. We assess the method for a comprehensive set of examples, relevant for predicting mechanical properties, such as yield strength or ductility, including dislocation bow-out, dislocation-precipitate interaction, and dislocation cross-slip. The main result of our analysis is that the FBC method is robust, easy-to-use, and up to two orders of magnitude more efficient than the current state-of-the-art method for this class of problems, the periodic array of dislocations (PAD) method, in terms of the required number of per-atom force computations when both methods give similar accuracy. This opens new prospects for large-scale atomistic simulations — without having to worry about spurious image effects that plague classical boundary conditions.

Recently, physics-driven deep learning methods have shown particular promise for the prediction of physical fields, especially to reduce the dependency on large amounts of pre-computed training data. In this work, we target the physics-driven learning of complex flow fields with high resolutions. We propose the use of Convolutional neural networks (CNN) based U-net architectures to efficiently represent and reconstruct the input and output fields, respectively. By introducing Navier-Stokes equations and boundary conditions into loss functions, the physics-driven CNN is designed to predict corresponding steady flow fields directly. In particular, this prevents many of the difficulties associated with approaches employing fully connected neural networks. Several numerical experiments are conducted to investigate the behavior of the CNN approach, and the results indicate that a first-order accuracy has been achieved. Specifically for the case of a flow around a cylinder, different flow regimes can be learned and the adhered “twin-vortices” are predicted correctly. The numerical results also show that the training for multiple cases is accelerated significantly, especially for the difficult cases at low Reynolds numbers, and when limited reference solutions are used as supplementary learning targets.

The paper presents a novel pressure-corrected formulation of the immersed boundary method (IBM) for the simulation of fully compressible non-Boussinesq natural convection flows. The formulation incorporated into the pressure-based fractional step approach facilitates simulation of the flows in the presence of an immersed body characterized by a complex geometry. Here, we first present extensive grid independence and verification studies addressing incompressible pressure-driven flow in an extended channel and non-Boussinesq natural convection flow in a differentially heated cavity. Next, the steady-state non-Boussinesq natural convection flow developing in the presence of hot cylinders of various diameters placed within a cold square cavity is thoroughly investigated. The obtained results are presented and analyzed in terms of the spatial distribution of path lines and temperature fields and of heat flux values typical of the hot cylinder and the cold cavity surfaces. Flow characteristics of multiple steady-state solutions discovered for several configurations are presented and discussed in detail.

Many configurations in plasma physics are axisymmetric, it will be more convenient to depict them in cylindrical coordinates compared with Cartesian coordinates. In this paper, a gas-kinetic scheme for collisional Vlasov-Poisson equations in cylindrical coordinates is proposed, our algorithm is based on Strang splitting. The equation is divided into two parts, one is the kinetic transport-collision part solved by multiscale gas-kinetic scheme, and the other is the acceleration part solved by a Runge-Kutta solver. The asymptotic preserving property of whole algorithm is proved and it’s applied on the study of charge separation problem in plasma edge and 1D Z-pinch configuration. Numerical results show it can capture the process from non-equilibrium to equilibrium state by Coulomb collisions, and numerical accuracy is obtained.

When two distinct ordered phases contact, the interface may exhibit rich and fascinating structures. Focusing on the Cylinder-Gyroid interface system, transition pathways connecting various interface morphologies are studied armed with the Landau–Brazovskii model. Specifically, minimum energy paths are obtained by computing transition states with the saddle dynamics. We present four primary transition pathways connecting different local minima, representing four different mechanisms of the formation of the Cylinder-Gyroid interface. The connection of Cylinder and Gyroid can be either direct or indirect via Fddd with three different orientations. Under different displacements, each of the four pathways may have the lowest energy.

We concentrate on the parallel, fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetry-preserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions. In this article, motivated by [M. J. Gander, S. Loisel, D. B. Szyld, SIAM J. Matrix Anal. Appl. 33 (2012) 653–680] and [S. Nardean, M. Ferronato, A. S. Abushaikha, J. Comput. Phys. 442 (2021) 110513], we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain, and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems. Robustness, computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.

In this work, we develop a novel high-order discontinuous Galerkin (DG) method for solving the incompressible Navier-Stokes equations with variable density. The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term. Then, since the hyperbolic nature of the governing equations is recovered, the simple and robust Harten-Lax-van Leer (HLL) flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations. The viscous term is discretized by the direct DG (DDG) method, the construction of which was initially inspired by the weak solution of a scalar diffusion equation. In addition, in order to eliminate the spurious oscillations around sharp density gradients, a local slope limiting operator is also applied during the highly stratified flow simulations. The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases. Due to its advantages of simplicity and robustness in implementation, the present method offers an effective approach for simulating the variable density incompressible flows.

A well-balanced second order finite volume central scheme for the magnetohydrodynamic (MHD) equations with gravitational source term is developed in this paper. The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells. A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme. The divergence-free constraint of the magnetic field is satisfied after applying the constrained transport method (CTM) for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field. The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.