The decoupled subbands model is widely used to approximate the two-dimensional Schrödinger equation. While computationally efficient, this simplified model loses accuracy in complex quantum transport simulations because it omits inter-subband interactions. We propose a novel transfer matrix method based on the multi-subband framework to solve the two-dimensional Schrödinger equation, which explicitly incorporates interactions among low-energy subbands. The transfer matrix construction leads to an overdetermined system, so we solve it using either least-squares fitting or singular value decomposition. We assess the accuracy of the proposed method with numerical experiments, showing that both approaches can achieve an error of $\mathcal{O}(\Delta x + \Delta y^2)$. To demonstrate practical utility, we simulate the current-voltage characteristics of resonant tunneling diodes, confirming the method's reliability for realistic quantum device simulations. \end{abstract}

In the present paper numerical solution of the nonlinear coupled system of sine-Gordon equations is studied in the weak sense. A first-order accurate and two second-order accurate unconditionally stable difference schemes corresponding to the system of sine-Gordon equations are considered. Solutions of these difference schemes are presented in the space of distributions by variational methods. The fixed point theory and the finite difference method are combined in the numerical implementations, carried out in MATLAB, in order to verify the theoretical results.

In this study, we implement a Crank-Nicolson finite difference scheme to discretize the (2+1)D saturable nonlinear Schrödinger equation with general damping. We show the existence and uniqueness of the discrete solution. The boundedness of the discrete mass and energy is established. The error between the exact and discrete solutions is shown to converge at a second-order rate in both time and space, according to the $L^2$ and $H^1$ discrete norms. Moreover, we show that the proposed scheme preserves the mass conservation and energy conservation for the (2+1)D saturable nonlinear Schrödinger equation without damping. Numerical simulations are conducted to validate these convergence properties and the conservation laws.

In this paper, the Cauchy problem of the nonlinear Helmholtz-type equation is discussed. This problem is well known to be severely ill-posed. Compared with linear equations, nonlinear equations are more difficult to deal with because of their lack of linearity. In order to obtain the approximate solution of this nonlinear problem, a combination of the quasi-boundary value method and the quasi-reversibility method is proposed. Using the variable separation method, the approximate solution is equivalent to solving a class of integral equations. The well-posedness of the approximation problem is proved by the Banach fixed-point theorem. Convergence analysis and error estimation are discussed. Since the convergence of error estimates cannot be obtained by the traditional a-priori bound, we introduce a new a-priori bound to obtain the convergence of error estimates, and the Hölder-type error estimate is achieved. Finally, some numerical experiments are given to corroborate the qualitative analysis and show the regularization method works well.

This study improves understanding of the interaction between magnetic fields and natural convection in complex geometries, addressing a significant challenge in engineering design. By analyzing the effects of multidirectional magnetic fields on fluid flow and heat transfer, this study provides valuable insights into optimizing thermal management in practical applications, such as exhaust manifolds. This work is motivated by the need to enhance thermal efficiency and control in systems with irregular geometries, which are prevalent in advanced technological applications. Employing magnetohydrodynamic (MHD) principles, the flow and heat exchange dynamics are analyzed under real-world conditions, considering a cavity of length $L$ with heated bottom walls and heated/cooled inner obstacles. The circular cavity contains heated rods with different radii. An inclined magnetic field, perpendicular to the fluid flow, is applied, while a no-slip condition is enforced at the walls. Finite Element Method simulations are conducted over a wide range of Rayleigh ($Ra$) and Hartmann ($Ha$) numbers with a fixed Prandtl number ($Pr=6.2$). The streamlines, isotherms, and two-dimensional (2D) plots are produced to show the effects of governing parameters. The present results are compared with the existing data and graphical results. The numerical results reveal that heat transfer is influenced not only by Rayleigh's number and magnetic field strength but also by the magnetic field's inclination.

Image smoothing techniques are widely used in computer vision and graphics applications. In this paper, we present an $\ell_p$-nonconvex regularization method for image smoothing. To induce sparsity prior of the smoothed images more strongly than the $\ell_1$ norm regularization, we take the nonconvex arctangent penalty function of the image gradient as the regularization term. To make the proposed model more flexible and effective for different image smoothing applications, we use the $\ell_p$ norm function as the fidelity term, instead of the $\ell_2$ norm function. The powerful majorization-minimization (MM) algorithm is employed for the proposed nonconvex optimization model. The convergence of the resulting MM algorithm is discussed. Comprehensive experiments and comparisons show that the proposed method is effective in various image processing tasks such as texture smoothing, detail enhancement, artifact removal, image denoising, high dynamic range (HDR) tone mapping, edge detection, and image composition.

In this paper, we propose two first-order, unconditionally stable, fully decoupled methods based on the Gauge-Uzawa method for solving the penalty Ericksen-Leslie system with variable density. The first method is based on the scalar auxiliary variable method for explicitly handling the nonlinear part, and the Gauge-Uzawa method for decoupling the velocity and pressure. The second method is formulated by introducing an auxiliary intermediate velocity variable for decoupling the director field computation from the velocity field, and then combining it with the Gauge-Uzawa method and the convex splitting method. Thus, two highly efficient, fully decoupled schemes are built. Furthermore, we use the finite element method to give an efficient implementation of the schemes. Then, we prove that both two schemes are unconditionally stable. Finally, numerical examples are provided to verify the convergence rate, unconditional stability, and computational efficiency. The proposed methods also provide a solution for establishing numerical schemes for other highly nonlinear variable-density coupled systems.

In this study, a class of explicit structure-preserving Du Fort-Frankel-type FDMs are firstly developed for Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation. They inherit some properties of the continuous problems, such as non-negativity, maximum principle and monotonicity. Besides, by using the discrete maximum principle, the error estimate in $L^\infty$-norm is proven to be $O\left(\tau + h_x^2 + h_y^2 + \left(\frac{h_x}{\tau}\right)^2 + \left(\frac{h_y}{\tau}\right)^2\right)$ as some suitable conditions are satisfied. Here, $\tau$, $h_x$ and $h_y$ are time step and spatial meshsizes in $x$- and $y$- directions, respectively. Then, as the current FDMs are used to solve Allen-Cahn equation, the obtained numerical solutions satisfy the discrete maximum principle and the discrete energy-dissipation law. Our methods are easy to be implemented because of explicitness. Finally, numerical results confirm theoretical findings and the efficiency of our methods.

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new $Log$ marking strategy is proposed. The $Log$ marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes.