Applied Numerical Mathematics最新文献

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.

In this paper, we investigate the implementation of the finite difference method on arbitrary meshes in conjunction with a fixed-point algorithm for the lubrication problem of a journal bearing with cavitation, considering the Elrod-Adams model. We establish numerical properties of the generalized finite difference scheme and provide several illustrative examples.

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of $n\ge 2$ suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete $L^2$-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete $L^2$-norm and $H^1$-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.

We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order $1-\alpha$, $\alpha \in (0,1)$, on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree $\beta \in (1,2]$. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative.

In this paper, the multinode Shepard method is adopted for the first time to numerically solve a differential problem with a discontinuity in the boundary. Starting from previous studies on elliptic boundary value problems, here the Shepard method is employed to catch the singularity on the boundary. Enrichments of the functional space spanned by the multinode cardinal Shepard basis functions are proposed to overcome the difficulties encountered. The Motz's problem is considered as numerical benchmark to assess the method. Numerical results are presented to show the effectiveness of the proposed approach.

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0\in H_0^1\left(\Omega \right)\cap H^2\left(\Omega \right)$. The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in $H^1$-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an $L^{\infty }$ error estimate is obtained for 2D problems that too with the initial condition is in $H_0^1\left(\Omega \right)\cap H^2\left(\Omega \right)$. All results proved in this paper are valid uniformly as $\alpha \to 1^{-}$, where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class.

In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes.