查看更多>>摘要：We introduce a family of symmetric stochastic Bernstein polynomials based on order statistics, and establish the order of convergence in probability in terms of the second order Ditzian-Totik type modulus of smoothness on the interval [0, 1], which epitomizes an optimal pointwise error estimate for the classical Bernstein polynomial approximation. Monte Carlo simulation results (presented at the end of the article) show that this new approximation scheme is efficient and robust. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this work, we design a numerically efficient finite-difference technique for the solution of a fractional extension of the Higgs boson equation in the de Sitter space- time. The model under investigation is a multidimensional equation with Riesz fractional derivatives of orders in (0, 1) boolean OR (1, 2], which considers a generalized potential and a time-dependent diffusion factor. An energy integral for the mathematical model is readily available, and we propose an explicit and consistent numerical technique based on fractional-order centered differences with similar Hamiltonian properties as the continuous model. A fractional energy approach is used then to prove the properties of stability and convergence of the technique. For simulation purposes, we consider both the classical and the fractional Higgs real-valued scalar fields in the (3 + 1)-dimensional de Sitter space-time, and find results qualitatively similar to those available in the literature. The present work is the first paper to report on a Hamiltonian discretization of the Higgs boson equation (both fractional and non-fractional) in the de Sitter space-time and its numerical analysis. More precisely, the present manuscript is the first paper of the literature in which a dissipation-preserving scheme to solve the multi-dimensional (fractional) Higgs boson equation in the de Sitter space-time is proposed and thoroughly analyzed. Indeed, it is worth pointing out that previous efforts used techniques based on the Runge-Kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed. (c) 2020 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we construct efficient collocation methods to deal with linear Volterra integral equations with highly oscillatory kernel. To discretize the oscillatory integrals in the collocation equation, a Filon type method is adopted. Based on some lemmas, we analyze the asymptotic property of the solution and obtain the convergence of the method, which is related to both the frequency omega and step length h. The proposed method does not only have the same order at the collocation points as the classical collocation method, but also may enjoy an asymptotic order which could reach two in some cases. Some numerical examples are presented at last to show the theoretical results and the efficiency of the method. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we use maximum principle preserving (MPP) and positivity-preserving (PP) parametrized flux limiters to achieve strict maximum principle and positivity-preserving property for the high order spectral volume (SV) scheme for solving hyperbolic conservation laws. This research is based on a generalization of the MPP and PP parametrized flux limiters in Xu (2014) and Christlieb et al. (2015) with Runge-Kutta (RK) time discretizations. For constructing MPP (PP) RK-SV schemes for hyperbolic conservation laws, we first focus on the RK-SV schemes to discuss how to apply MPP or PP parametrized flux limiters. Then we design and analyze high order MPP RK-SV schemes for scalar conservation laws, and high order PP RK-SV schemes for compressible Euler systems. The efficiency and effectiveness of the proposed schemes are demonstrated via a set of numerical experiments. Both the analysis and numerical experiments indicate that the proposed scheme without any additional time step restriction, not only preserves the maximum principle of the numerical approximation, but also maintains the designed high-order accuracy of the SV scheme for linear advection problems. (C) 2021 Published by Elsevier B.V.
查看更多>>摘要：This paper summarizes the results of a numerical investigation of the phenomenon of escape in the N-body ring problem. There is a value of the Jacobi constant of the system such that for smaller values, the potential well opens and test particles may leave the potential through any of its N openings. By means of a surface of section, we show the results of the computation of the basins of escape towards the different directions for N = 5, 6, 7, 8. We have also analyzed the percentage of escaping orbits, the direction of escape and the distribution of the times of escape.(C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper we present a novel framework for obtaining high-order numerical methods for scalar conservation laws in one-space dimension for both the homogeneous and nonhomogeneous cases (or balance laws). The numerical schemes for these two settings are somewhat different in the presence of shocks, however at their core they both rely heavily on the solution curve being represented parametrically. By utilizing highorder parametric interpolation techniques we succeed to obtain fifth order accuracy (in space) everywhere in the computation domain, including the shock location itself. In the presence of source terms a slight modification is required, yet the spatial order is maintained but with an additional temporal error appearing. We provide a detailed discussion of a sample scheme for non-homogeneous problems which obtains fifth order in space and fourth order in time even in the presence of shocks. (C) 2021 Elsevier B.V. All rights reserved.
Ezquerro, J. A.Hernandez-Veron, M. A.Magrenan, a. A.
11页
查看更多>>摘要：We establish a global convergence result for an efficient third-order iterative process which is constructed from Chebyshev's method by approximating the second derivative of the operator involved by combinations of the operator. In particular, from the use of auxiliary points, we provide domains of restricted global convergence that allow obtaining balls of convergence and locate solutions. Finally, we use different numerical examples, including a Chandrashekar's integral equation problem, to illustrate the study. (C) 2021 The Authors. Published by Elsevier B.V.
查看更多>>摘要：In this paper, we consider a class of parameterized singularly perturbed problems with integral boundary condition. A finite difference scheme of hybrid type with an appropriate Shishkin mesh is suggested to solve the problem. We prove that the method is of almost second order convergent in the discrete maximum norm. Numerical results are presented, which illustrate the theoretical results. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we investigate the stability of a numerical method for solving the wave equation. The method uses explicit leap-frog in time and high order continuous and discontinuous (DG) finite elements using the standard Lagrange and Hermite basis functions in space. Matrix eigenvalue analysis is used to calculate time-step restrictions. We show that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, such as equidistributed Lagrange nodes and Gauss-Lobatto nodes. We show that the time-step restriction for the symmetric interior penalty DG schemes with the usual penalty terms is tighter than for continuous Lagrange finite elements. Finally, we conclude that the best time-step restriction is obtained for continuous Hermite finite elements up to polynomial degrees p = 13. (C) 2021 The Author(s). Published by Elsevier B.V.
查看更多>>摘要：We consider a space-time adaptive splitting scheme for polycrystallization processes described by a two-field phase field model. The phase field model consists of a coupled system of evolutionary processes for the local degree of crystallinity phi and the orientation angle Theta one of them being of first order total variation flow type. The splitting scheme is based on an implicit discretization in time which allows a decoupling of the system in the sense that at each time step minimization problems in phi and Theta have to be solved successively. The discretization in space is taken care of by a standard finite element approximation for the problem in phi and an Interior Penalty Discontinuous Galerkin (IPDG) approximation for the one in Theta. The adaptivity in space relies on equilibrated a posteriori error estimators for the discretization errors in phi and Theta in terms of primal and dual energy functionals associated with the respective minimization problems. The adaptive time stepping is dictated by the convergence of a semismooth Newton method for the numerical solution of the nonlinear problem in Theta. Numerical results illustrate the performance of the adaptive space-time splitting scheme for two representative polycrystallization processes. (C) 2021 Elsevier B.V. All rights reserved.
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