Applied Numerical Mathematics最新文献

This paper is dedicated to the numerical solution of a mathematical model that describes frictional quasistatic contact between an elastic body and a moving foundation, with the wear effect on the contact interface of the moving foundation due to friction. The mathematical problem is a system consisting of a time-dependent quasi-variational inequality and an integral equation. The numerical method is based on the use of the virtual element method (VEM) for the spatial discretization of the variational inequality and a variable step-size left rectangle integration formula for the integral equation. The existence and uniqueness of a numerical solution are shown, and optimal order error estimates are derived for both the displacement and the wear function for the lowest order VEM. Numerical results are presented to demonstrate the efficiency of the method and to illustrate the numerical convergence orders.

This article focuses on developing and analyzing an efficient higher-order numerical approximation of singularly perturbed two-dimensional semilinear parabolic convection-diffusion problems with time-dependent boundary conditions. We approximate the governing nonlinear problem by an implicit fitted mesh method (FMM), which combines an alternating direction implicit scheme in the temporal direction together with a higher-order finite difference scheme in the spatial directions. Since the solution possesses exponential boundary layers, a Cartesian product of piecewise-uniform Shishkin meshes is used to discretize in space. To begin our analysis, we establish the stability corresponding to the continuous nonlinear problem, and obtain a-priori bounds for the solution derivatives. Thereafter, we pursue the stability analysis of the discrete problem, and prove ε-uniform convergence in the maximum-norm. Next, for enhancement of the temporal accuracy, we use the Richardson extrapolation technique solely in the temporal direction. In addition, we investigate the order reduction phenomenon naturally occurring due to the time-dependent boundary data and propose a suitable approximation to tackle this effect. Finally, we present the computational results to validate the theoretical estimates.

A modified Polak Ribiere Polyak(PRP) conjugate gradient(CG) method is proposed for solving unconstrained optimization problems. The search direction generated by this method satisfies sufficient descent condition at each iteration and this method inherits one remarkable property of the standard PRP method. Under the standard Armijo line search, the global convergence and the linearly convergent rate of the presented method is established. Some numerical results are given to show the effectiveness of the proposed method by comparing with some existing CG methods.

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

This paper mainly presents numerical solutions to an initial-boundary value problem of the time-fractional Klein-Gordon equations. We developed a numerical scheme with the help of the finite difference methods and the predictor-corrector methods to find numerical solutions of the considered problems. The proposed scheme is based on discretizing the considered problems with respect to spatial and temporal domains. Numerical results are derived for some illustrative problems, and the outputs are compared with the exact solution in the integer order case. The solution behavior and 3D graphics of the discussed problems are demonstrated using the proposed scheme. Finally, the proposed scheme, which does not require solving large systems of linear equations, can be extended and modified to handle other classes of time-fractional PDEs.

In this paper, we consider the graph regularization Q-weighted nonnegative matrix factorization problem in multi-view clustering. Based on the Q-weighted norm property, this problem is transformed into the minimization problem of the trace function. The necessary condition for the existence of a solution is given. The proximal alternating nonnegative least squares method and its acceleration method are designed to solve it. The convergence theorem is also given. The feasibility and effectiveness of the proposed methods are verified by numerical experiments.

In this study, we introduce a multi-step multi-derivative predictor-corrector time integration scheme analogous to the schemes in Schütz et al. (2022) [13], incorporating a multi-step quadrature rule. We conduct stability analysis up to order eight and optimize the schemes to achieve $A\left(\alpha \right)$-stability for large α. Numerical experiments are performed on ordinary differential equations exhibiting diverse stiffness conditions, as well as on partial differential equations showcasing non-linearity and higher-order terms. Results demonstrate the convergence and flexibility of the proposed schemes across diverse situations.

In this paper, we introduce a velocity correction projection method for the Micropolar Navier-Stokes Equations. The velocity correction method are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. At the same time, the optimal error estimate using the technique of dual norm are obtained. In this way, the divergence free of the velocity u can be conserved. Finally, the numerical results show the method has an optimal convergence order. The numerical results are consistent with our theoretical analysis, and our method is effective.

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis.

The non-equilibrium structural and dynamical properties of a flexible polymer tethered to a reflecting wall and subject to oscillatory linear flow are studied by numerical simulations. Polymer is confined in two dimensions and is modeled as a bead-spring chain immersed in a fluid described by the Brownian multiparticle collision dynamics. At high strain, the polymer is stretched along the flow direction following the applied flow, then recoils at flow inversion before flipping and elongate again. When strain is reduced, it may happen that the chain recoils without flipping when the applied shear changes sign. Conformations are analyzed and compared to stiff polymers revealing more compact patterns at low strains and less stretched configurations at high strain. The dynamics is investigated by looking at the center-of-mass motion which shows a frequency doubling along the direction normal to the external flow. The center-of-mass correlation function is characterized by smaller amplitudes when reducing bending rigidity.