Phase-field models are widely used in studying multiphase flow dynamics. Given the complexity and strong nonlinearity, designing accurate, efficient, and stable numerical algorithms to solve these models has been an active research field for decades. This paper proposes a novel numerical scheme to solve a highly cited and used phase field hydrodynamic model for simulating ternary phase fluid flows. The main novelty is the introduction of a supplementary variable to reformulate the original problem into a constrained optimization problem. This reformulation leads to several advantages for our proposed numerical algorithms compared with many existing numerical techniques for solving this model. First, the developed schemes allow more straightforward calculations for the hydrodynamic phase-field models by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix, remarkably reducing the computational cost. Secondly, the numerical schemes can maintain mass conservation and energy dissipation at the discrete level. Additionally, the developed scheme based on the second-order backward difference formula respects the original energy dissipation law that differs from many existing schemes, such as the IEQ, SAV, and Lagrange multiplier approaches for which a modified energy dissipation law is respected. Furthermore, rigorous proof of energy stability and practical implementation strategies are provided. We conduct adequate 2D and 3D numerical tests to demonstrate the proposed schemes’ accuracy and effectiveness.

In [Dai et al., Multi. Model. Simul. 18(4) (2020)], a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory, based on which a linearized method was developed in [Hu et al., EAJAM. 13(2) (2023)] for further improving the numerical efficiency. In this paper, a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model. Temporally, the convergence, the asymptotic stability, as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works, while spatially, the convergence of the $h$-adaptive mesh method is demonstrated following [Chen et al., Multi. Model. Simul. 12 (2014)], with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham model. Numerical examples confirm the theoretical results very well.

A novel dynamical model with fixed-time convergence is presented to solve the system of absolute value equations (AVEs). Under a mild condition, it is proved that the solution of the proposed dynamical system converges to the solution of the AVEs. Moreover, in contrast to the existing inversion-free dynamical system (C. Chen et al., Appl. Numer. Math. 168 (2021), 170–181), a conservative settling-time of the proposed method is given. Numerical simulations illustrate the effectiveness of the new method.

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the second-order accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.

Deep learning has achieved great success in solving partial differential equations (PDEs), where the loss is often defined as an integral. The accuracy and efficiency of these algorithms depend greatly on the quadrature method. We propose to apply quasi-Monte Carlo (QMC) methods to the Deep Ritz Method (DRM) for solving the Neumann problems for the Poisson equation and the static Schrödinger equation. For error estimation, we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error, the approximation error and the training error. We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM. Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM, which illustrates the superiority of QMC in deep learning over MC.

The rigorous relations between the propagators of transient Schrödinger equations and stationary Green functions are established. Based on the generalized Fourier transform, non-singular transparent boundary condition for transient problem is proposed in a representation of Green functions. The unified framework of Green function method is presented for converting an open boundary problem into a bounded boundary problem. Numerical scheme for time-dependent Schrödinger equation with non-singular transparent boundary condition is designed to simulate the propagations of a free Gaussian wave packet and the resonant tunnelling through double barriers. Numerical results validate the effectiveness of non-singular transparent boundary condition.

Character models have enormous applications in industry. Efficient creation of detailed character models is an important topic. This paper proposes a new and easy-to-use technique to quickly create detailed character models from sketches. The proposed technique consists of two main components: primitive deformer and shape generators. With this technique, 2D silhouette contours of a character model are drawn or extracted from an image or sketch. Then, proper geometric primitives are selected and aligned with the corresponding 2D silhouette contours. After that, a primitive deformer is used to create a base mesh and three shape generators are used to add 3D details to the base mesh. The primitive deformer and three shape generators are developed from ODE-driven deformations. The primitive deformer deforms the aligned geometric primitives to exactly match the 2D silhouette contours in one view plane and obtains a base mesh of a character model consisting of deformed primitives. The shape generators are used to add 3D details to the base mesh by creating local 3D models. The experimental results demonstrate that the new technique can quickly create detailed 3D character models from sketches with few manual operations. The new technique is physics-based and easy to learn and use.

Two mathematical models in the context of boundary value problems are proposed for the geometric design of letters in Times Roman font. We adopt radial basis function meshless collocation method for numerically solving the two proposed mathematical models in 2D and 3D. In this paper, Bézier curves play an important role in the design of the letters. Three examples with simply and multiply-connected domains in 2D and 3D are presented to demonstrate the visual effect of the letters in Times Roman font.

We propose Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which are a generalization of MC-fPINNs in L. Guo et al. (Comput. Methods Appl. Mech. Eng. 400 (2022), 115523) for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar to MC-fPINNs, our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach.

In this work, we consider a combined finite element method for fully coupled nonlinear thermo-poroelastic model problems. The mixed finite element (MFE) method is used for the pressure, the characteristics finite element (CFE) method is used for the temperature, and the Galerkin finite element (GFE) method is used for the elastic displacement. The semi-discrete and fully discrete finite element schemes are established and the stability of this method is presented. We derive error estimates for the pressure, temperature and displacement. Several numerical examples are presented to confirm the accuracy of the method.

We propose an alternating direction method of multipliers (ADMM) to solve an optimization problem stemming from inverse lithography. The objective functional of the optimization problem includes three terms: the misfit between the imaging on wafer and the target pattern, the penalty term which ensures the mask is binary and the total variation regularization term. By variable splitting, we introduce an augmented Lagrangian for the original objective functional. In the framework of ADMM method, the optimization problem is divided into several subproblems. Each of the subproblems can be solved efficiently. We give the convergence analysis of the proposed method. Specially, instead of solving the subproblem concerning sigmoid, we solve directly the threshold truncation imaging function which can be solved analytically. We also provide many numerical examples to illustrate the effectiveness of the method.