Calcolo最新文献

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order (alpha in (1,2).) The error has the asymptotic expansion ( big ( d_{3} tau ^{3- alpha } + d_{4} tau ^{4-alpha } + d_{5} tau ^{5-alpha } + cdots big ) + big ( d_{2}^{*} tau ^{4} + d_{3}^{*} tau ^{6} + d_{4}^{*} tau ^{8} + cdots big ) ) at any fixed time (t_{N}= T, N in {mathbb {Z}}^{+}), where (d_{i}, i=3, 4,ldots ) and (d_{i}^{*}, i=2, 3,ldots ) denote some suitable constants and (tau = T/N) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order (alpha in (1,2)) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

We propose two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral (SPEWIFP) methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. In this regime, the solution of Dirac equation is highly oscillatory in time because of the small parameter (0 <varepsilon ll 1) which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition (tau lesssim 1) and preserve the modified energy and modified mass in the discrete level. Although our methods can only preserve the modified energy and modified mass instead of the original energy and mass, our methods are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are often implicit. Through rigorous error analysis, we give the error bounds of the methods at (O(h^{m_0} + tau ^2/varepsilon ^2)) where h is mesh size, (tau ) is time step and the integer (m_0) is determined by the regularity conditions. These error bounds indicate that, to obtain the correct numerical solution in the simultaneously massless and nonrelativistic regime, our methods request the (varepsilon )-scalability as (h = O(1)) and (tau = O(varepsilon )) which is better than the (varepsilon )-scalability of the finite difference (FD) methods: (h =O(varepsilon ^{1/2})) and (tau = O(varepsilon ^{3/2})). Numerical experiments confirm that the theoretical results in this paper are correct.

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings.

In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal (H^1) error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions.

Elliptic polytopes are convex hulls of several concentric plane ellipses in ({{mathbb {R}}}^d). They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for (d=2, 3), it admits an explicit solution. For larger d, two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.

Embedding randomization procedures in the Alternating Direction Method of Multipliers (ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in practice, the introduction of such techniques could mitigate the diverging behaviour of the multi-block extension of ADMM, from the theoretical point of view, it can ensure just the convergence in expectation, which may not be a good indicator of its robustness and efficiency. In this work, analysing the strongly convex quadratic programming case from a linear algebra perspective, we interpret the block Gauss–Seidel sweep performed by the multi-block ADMM in the context of the inexact Augmented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative technique to those present in the literature which, supported from stronger theoretical guarantees, is able to ensure the convergence of the multi-block generalization of the ADMM method.

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only ({mathcal {O}}(N)) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions.