查看更多>>摘要：A class of high order extended boundary value methods (HEBVMs) suitable for the numerical approximation of stiff systems of ordinary differential equations (ODEs) is constructed. This class of BVMs is based on the second derivative class of linear multistep formulas (LMF) and it provides a set of very highly stable methods that can produce considerably accurate solutions to stiff systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The class of BVMs derived herein is of high order, small error constants and large region of absolute stability. Specifically, it is O-k1,O- k2-stable, A(k1, k2)-stable with (k(1), k(2))-boundary conditions and order p = k + 4 for values of the step length k >= 1. The numerical results obtained from standard linear and non-linear stiff systems indicate that this scheme is highly competitive with existing methods. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we study the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The compact finite difference schemes could achieve very high order accuracy with relatively small stencils. To match the convergence order of the compact schemes in the interior domain, we take the simplified inverse Lax-Wendroff (SILW) procedure (Tan et al., 2012; Li et al., 2017) as our numerical boundary treatment. The third order total variation diminishing (TVD) Runge-Kutta method (Shu and Osher, 1988) is taken as our time-stepping method in the fully-discrete case. Two analysis techniques are adopted to check the algorithm's stability, one is based on the Godunov-Ryabenkii theory, and the other is the eigenvalue spectrum visualization method (Vilar and Shu, 2015). Both the semi-discrete and fully-discrete cases are investigated, and these two different analysis techniques yield consistent results. Several numerical experimental results are shown to validate the theoretical results. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Multi-degree splines are piecewise polynomial functions having sections of different degrees. They offer significant advantages over the classical uniform-degree framework, as they allow for modeling complex geometries with fewer degrees of freedom and, at the same time, for a more efficient engineering analysis. Moreover they possess a set of basis functions with similar properties to standard B-splines. In this paper we develop an algorithm for efficient evaluation of multi-degree B-splines, which, unlike previous approaches, is numerically stable. The proposed method consists in explicitly constructing a mapping between a known basis and the multi-degree B-spline basis of the space of interest, exploiting the fact that the two bases are related by a sequence of knot insertion and/or degree elevation steps and performing only numerically stable operations. In addition to theoretically justifying the stability of the algorithm, we will illustrate its performance through numerical experiments that will serve us to demonstrate its excellent behavior in comparison with existing methods, which, in some cases, suffer from apparent numerical problems. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper presents the rates of uniform strong consistency of wavelet estimation for nonparametric function in sup-norm loss by introducing an empirical process approach. A compact support assumption on the explanatory variable is commonly used in nonparametric regression analysis. In the article, we consider the wavelet estimation analysis without any assumption on the compacity of the support of the explanatory variable. The optimal uniform convergence rates of the wavelet estimators are achieved by suitably choosing resolution level. These results are useful for wavelet theory on nonparametric signal recovery and analysis. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：A new heuristic parameter choice rule is proposed, which is an important process in solving the linear ill-posed integral equation. Based on multiscale Galerkin projection, we establish the error upper bound between the approximate solution obtained by this rule and the exact solution. Under certain conditions, we prove that the approximate solution obtained by this rule can reach the optimal convergence rate. Since the computational cost will be very large when the dimension of space increases, we analyze a special m-dimensional integral operator that can be transformed to m one-dimensional integral operator, which can reduce the computational cost greatly. Numerical experiments show that the proposed heuristic rule is promising among the known heuristic parameter choice rules. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：A new family of Explicit Runge-Kutta-Nystrom pair of orders seven and five is studied here. Its main advantage is that it spends only six stages per step. This is a remarkable improvement since only pairs of orders 6(4) were attained at this cost until now. We present a particular pair with minimal truncation error coefficients. Numerical results show the superiority of our proposal over a set of relevant problems. (C) 2021 Elsevier B.V. All rights reserved.
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