We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

We propose in this work a second-order Langevin sampler for the isothermal-isobaric ensemble (the NPT ensemble), preserving a positive volume for the simulation box. We first derive the suitable equations of motion for particles to be coupled with the overdamped Langevin equation of volume by sending the artificial mass of the periodic box to zero in the work of Liang et al., [J. Chem. Phys., 157 (14)]. We prove the well-posedness of the new system of equations and show that its invariant measure is the desired ensemble. The new continuous time equations not only justify the previous cell rescaling methods, but also allow us to choose a suitable friction coefficient so that one has additive noise after a change of variable by taking logarithm of the volume. This observation allows us to propose a second-order weak scheme that guarantees the positivity of the volume. Various numerical experiments have been performed to demonstrate the efficacy of our method.

In this paper, we present a neural network approach to address the dynamic unbalanced optimal transport problem on surfaces with point cloud representation. For surfaces with point cloud representation, traditional method is difficult to apply due to the difficulty of mesh generating. Neural network is easy to implement even for complicate geometry. Moreover, instead of solving the original dynamic formulation, we consider the Hamiltonian flow approach, i.e. Karush-Kuhn-Tucker system. Based on this approach, we can exploit mathematical structure of the optimal transport to construct the neural network and the loss function can be simplified. Extensive numerical experiments are conducted for surfaces with different geometry. We also test the method for point cloud with noise, which shows stability of this method. This method is also easy to generalize to diverse range of problems.

The supplementary variable method has received much attention as a new technique to construct structure-preserving algorithm that can preserve the original energy law. However, there has been a lack of relevant convergence analysis results, mainly because the method requires solving a nonlinear algebraic equation for the supplementary variable, and the non-uniqueness of its root makes estimating the root’s error very difficult. In this paper, we take the Allen-Cahn model as an example and construct a second-order scheme using the supplementary variable method that preserves the original energy dissipation law. We then give a sufficient condition to ensure that the root of the nonlinear algebraic equation exists uniquely in the neighborhood of its exact solution. Under this condition, and when the time step is sufficiently small, we establish a rigorous error estimator for this scheme. Finally, we validate the effectiveness of the proposed scheme through several numerical examples.

We propose a novel approach to simulate the solution of the time-dependent Schrödinger equation with a general variable potential. The key idea is to approximate the Titchmarsh-Weyl $m$-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.

Currently existing energy-stable parametric finite element methods for surface diffusion flow and other flows are usually limited to first-order accuracy in time. Designing a high-order algorithm for geometric flows that can also be theoretically proven to be energy-stable poses a significant challenge. Motivated by the new scalar auxiliary variable approach [1], we propose novel energy-stable parametric finite element approximations for isotropic/anisotropic surface diffusion flows, achieving both first-order and second-order accuracy in time. Additionally, we apply the algorithms to simulate the solid-state dewetting of thin films. Finally, extensive numerical experiments validate the accuracy, energy stability, and efficiency of our developed numerical methods. The designed algorithms in this work exhibit strong versatility, as they can be readily extended to other high-order time discretization methods (e.g., BDFk schemes). Meanwhile, the algorithms achieve remarkable computational efficiency and maintain excellent mesh quality. More importantly, the algorithm can be theoretically proven to possess unconditional energy stability, with the energy nearly equal to the original energy.

The flux difference splitting (FDS) combined with the characteristic-wise weighted essentially non-oscillatory (WENO) reconstruction in single-component flow simulations is quite attractive and preferred for its better non-oscillatory feature near discontinuities. However, when compressible multi-component flows are simulated, the FDS with the characteristic-wise WENO reconstruction may cause spurious oscillations. In this work, we analyze the reason that the oscillations are generated. A new method, which can effectively suppress this kind of spurious oscillations in multi-component flows, is proposed by modifying the total energy in the process of the WENO reconstruction. For example, in the fifth-order WENO reconstruction, on the global stencil $S^5=(j−2,j−1,···,j+2)$ used to reconstruct the left-side value $U^L_{ j+1/2},$ the total energy values are first modified by using the reconstructed value of the specific heat ratio (i.e., $γ^L_{j+1/2},$ which can be directly reconstructed from the transport equation of the mass fraction), and then the characteristic-wise WENO reconstruction process is implemented. Numerical examples including several one-, two- and three-dimensional multi-component flows confirm the effectiveness and robustness of the proposed method.

The discrete direct deconvolution model (D3M) is developed for the large-eddy simulation (LES) of turbulence. The D3M is a discrete approximation of previous direct deconvolution model studied by Chang et al. ["The effect of sub-filter scale dynamics in large eddy simulation of turbulence," Phys. Fluids 34, 095104 (2022)]. For the first type model D3M-1, the original Gaussian and Helmholtz filters are approximated by local discrete formulation of different orders, and direct inverse of the discrete filter is applied to reconstruct the unfiltered flow field. The inverse of original Gaussian and Helmholtz filters can be also approximated by local discrete formulation, leading to a fully local model D3M-2. Compared to traditional models including the dynamic Smagorinsky model (DSM) and the dynamic mixed model (DMM), the D3M-1 and D3M-2 exhibit much larger correlation coefficients and smaller relative errors in the a priori studies. In the a posteriori validations, both D3M-1 and D3M-2 can accurately predict turbulence statistics, including velocity spectra, probability density functions (PDFs) of sub-filter scale (SFS) stresses and SFS energy flux, as well as time-evolving kinetic energy spectra, momentum thickness, and Reynolds stresses in turbulent mixing layer. D3M-1 and D3M-2 have more advantages in predicting the SFS statistics compared to scale-similarity model (SSM), DSM, and DMM. Thus, the D3M holds potential as an effective SFS modeling approach in turbulence simulations.

This paper focuses on modelling and numerical shape and topology optimization for a nonlinear hemivariational inequality in a Stokes flow with contact and frictional conditions. For the energy functional to be minimized subject to geometric volume constraint, we present shape sensitivity analysis and propose a volume preserving gradient-type method for shape optimization. Moreover, we build a relaxed topology optimization model for the hemivariational inequality in a Stokes fluid and solve it by a level set method. Numerical examples are presented to verify effectiveness of the shape and topology optimization algorithms.

A new discontinuous Galerkin (DG) method is proposed and analyzed for the Maxwell eigenproblem, featuring a local reconstruction of the curl operator in a discontinuous finite element space. The proposed method can be penalty-free or penalty-factor-free, depending on which discontinuous finite element space the curl operator is locally reconstructed in. The new DG method is recast into a saddle-point problem so that it can be analyzed from the Babuška-Osborn theory for the finite element approximation of the spectrum of the compact operator, and the convergence and the optimal error estimates are then obtained; the discrete eigenmodes are spurious-free and spectral-correct. We provide numerical results to illustrate the proposed method.