In recent years, phase-field-based models for multiphase flows have gained significant popularity, particularly within the lattice Boltzmann (LB) community. These models typically use two lattice Boltzmann equations (LBEs), one for interface tracking and the other for solving hydrodynamic properties. However, for the purposes of this paper, we focus only on the LB model for hydrodynamics. Our goal is to undertake a comparative investigation into the differences between three classical hydrodynamic LB models proposed by Lee et al. [1], Liang et al. [2] and Fakhari et al. [3]. The interface-tracking equation used in this study is based on the conservative phase-field model. We provide a detailed derivation of the governing equations in each model using the Chapman-Enskog analysis. Additionally, three discretization methods for the interaction forces are introduced, and a modified method for the gradient term is proposed based on the nonequilibrium distribution method. The accuracy of three LB models in combination with four discretization methods is examined in this study. Based on the results, it appears that different combinations of models and methods are appropriate for different types of problems. However, some suggestions for the selection of hydrodynamic models and discrete methods for the gradient term are provided in this paper.

We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method and results from the literature. The extreme fine-scale resolution features are illustrated showing the method’s capabilities to efficiently treat filamentation in fusion plasma simulations.

The development of high-order shock-capturing schemes is critical for compressible fluid simulations, in particular for cases where both shock waves and small-scale turbulence structures present. As one of the state-of-the-art high-order numerical schemes, the family of high-order targeted ENO (TENO) schemes proposed by Fu et al. [Journal of Computational Physics 305 (2016): 333-359] has been demonstrated to perform well for compressible gas dynamics on structured meshes and recently extended to unstructured meshes by Ji et al. [Journal of Scientific Computing 92(2022): 1-39]. In this paper, with the observation that the TENO scheme not only provides the high-order reconstructed data at the cell interface but also features the potential to separate the local flow scales in the wavenumber space, we propose a low-dissipation finite-volume TENO scheme with a new cell-interface flux evaluation strategy for unstructured meshes. The novelty originates from the fact that the local flow scales are classified, following a strong scale separation in the reconstruction process, as “very smooth” or not. When the corresponding large central-biased stencil for the targeted cell interface is judged to be “very smooth”, a low-dissipation Riemann solver, even the non-dissipative central flux scheme, is employed for the cell-interface flux computing. Otherwise, a dissipative approximate Riemann solver is employed to avoid spurious oscillations and achieve stable shock-capturing. Such a strategy provides separate control over the numerical dissipation of the high-order reconstruction process and the cell-interface flux calculation within a unified framework and leads to a resultant finite-volume method with extremely low-dissipation properties and good numerical robustness. Without parameter tuning case by case, a set of canonical benchmark simulations has been conducted to assess the performance of the proposed scheme.

In this study, we apply first-order exponential time differencing (ETD) methods to solve benchmark problems for the diffuse-interface model using the Peng-Robinson equation of state. We demonstrate the unconditional stability of the proposed algorithm within the ETD framework. Additionally, we analyzed the complexity of the algorithm, revealing that computations like matrix multiplications and inversions in each time step exhibit complexity strictly less than $\mathcal{O}(n^2),$ where $n$ represents the number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterative solutions (such as matrix inversion) in each time step, as they are sensitive to matrix properties. Instead, we adopted a hierarchical matrix ($\mathcal{H}$-matrix) approximation for the matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{\rm log}(n))$ for their product with an $n$-vector, which outperforms the traditional $\mathcal{O}(n^2)$ complexity. Overall, our focus is on creating an unconditionally stable algorithm with improved computational efficiency and reliability.

This paper proposes and analyzes a novel fully discrete finite element scheme with an interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. The nonlinear term satisfies a one-sided Lipschitz condition and the diffusion term is globally Lipschitz continuous. The novelties of this paper are threefold. Firstly, the $L^2$-stability ($L^∞$ in time) and $H^2$-stability ($L^2$ in time) are proved for the proposed scheme. The idea is to utilize the special structure of the matrix assembled by the nonlinear term. None of these stability results has been proved for the fully implicit scheme in existing literature due to the difficulty arising from the interaction of the nonlinearity and the multiplicative noise. Secondly, higher moment stability in $L^2$-norm of the discrete solution is established based on the previous stability results. Thirdly, the Hölder continuity in time for the strong solution is established under the minimum assumption of the strong solution. Based on these findings, the strong convergence in $H^{−1}$-norm of the discrete solution is discussed. Several numerical experiments including stability and convergence are also presented to validate our theoretical results.

In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave framework for the effective potentials of the incommensurate systems. And utilizing the effective potentials represented by the plane wave, the location of the electron density can be inferred. Moreover, the spectral distribution can be obtained from the effective potential version of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectral properties of the incommensurate systems, without solving the eigenvalue problem explicitly.

This study investigates high-order face and edge elements in finite element methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange finite elements, setting the foundation for further analysis. The discussion then extends to $H$(div)-conforming and $H$(curl)-conforming finite element spaces, adopting variable frames across differing sub-simplices. The imposition of tangential or normal continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and practice.

We propose a linear decoupled positivity-preserving scheme for the chemotaxis-fluid system, which models the interaction between aerobic bacteria and the fluid flow surrounding them. This scheme comprises the finite element method (FEM) for the fluid equations on a regular triangulation and an upwind finite volume method (FVM) for the chemotaxis system on two types of dual mesh. The discrete cellular density and chemical concentration are represented as the piecewise constant functions on the dual mesh. They can also be equivalently expressed as the piecewise linear functions on the triangulation in the sense of mass-lumping. These discrete solutions are obtained by the upwind finite volume approximation satisfying the laws of positivity preservation and mass conservation. The finite element method is used to compute the numerical velocity in the triangulation, which is then used to determine the upwind-style numerical flux in the dual mesh. We analyze the $M$-property of the matrices from the discrete system and prove the well-posedness and the positivity-preserving property. By using the $L^p$-estimate of the discrete Laplace operators, semigroup analysis, and induction method, we are able to establish the optimal error estimates for chemical concentration, cellular density, and velocity field in $(l^∞(W^{1,p}), l^∞(L^p),l^∞(W^{1,p}))$-norm. Several numerical examples are presented to verify the theoretical results.

With continuous developments in various techniques, machine learning is becoming increasingly viable and promising in the field of fluid mechanics. In this article, we present a machine learning approach for enhancing the resolution and robustness of the weighted compact nonlinear scheme (WCNS). We employ a neural network as a weighting function in the WCNS scheme and follow a data-driven approach to train this neural network. Neural networks can learn a new smoothness measure and calculate a weight function inherently. To facilitate the machine learning task and train with fewer data, we integrate the prior knowledge into the learning process, such as a Galilean invariant input layer and CNS polynomials. The normalization in the Delta layer (the so-called Delta layer is used to calculate input features) ensures that the WCNS3-NN schemes achieve a scale-invariant property (Si-property) with an arbitrary scale of a function, and an essentially non-oscillatory approximation of a discontinuous function (ENO-property). The Si-property and ENO-property of the data-driven WCNS schemes are validated numerically. Several one- and two-dimensional benchmark examples, including strong shocks and shock-density wave interactions, are presented to demonstrate the advantages of the proposed method.