Applied Numerical Mathematics最新文献

In this paper, by applying the supplementary variable method (SVM), some high-order, conformal structure-preserving, linearized algorithms are developed for the damped nonlinear Schrödinger equation. We derive the well-determined SVM systems with the conformal properties and they are then equivalent to nonlinear equality constrained optimization problems for computation. The deduced optimization models are discretized by using the Gauss type Runge-Kutta method and the prediction-correction technique in time as well as the Fourier pseudo-spectral method in space. Numerical results and some comparisons between this method and other reported methods are given to favor the suggested method in the overall performance. It is worthwhile to emphasize that the numerical strategy in this work could be extended to other conservative or dissipative system for designing high-order structure-preserving algorithms.

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

We develop a local discontinuous Galerkin (LDG) method for a fourth-order singularly perturbed problem of convection-diffusion type. The existence and uniqueness of the computed solution are verified. Using the Shishkin mesh we derive an optimal-order energy-norm error estimate which is uniformly valid in the perturbation parameter. Numerical experiments are also given to support our theoretical findings.

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.

In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.

Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.

The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

In this study, we introduce a novel hybrid conjugate gradient method with an adaptive strategy called asHCG method. The asHCG method exhibits the following characteristics. (i) Its search direction guarantees sufficient descent property without dependence on any line search. (ii) It possesses strong convergence for the uniformly convex function using a weak Wolfe line search, and under the same line search, it achieves global convergence for the general function. (iii) Employing the Armijo line search, it provides an approximate guarantee for worst-case complexity for the uniformly convex function. The numerical results demonstrate promising and encouraging performances in both unconstrained optimization problems and image restoration problems.

Investigation of the solutions of the coupled viscous Burgers system is crucial for realizing and understanding some physical phenomena in applied sciences. Particularly, Burgers equations are used in the modeling of fluid mechanics and nonlinear acoustics. In the present study, a modified meshless quadrature method based on radial basis functions is used to discretize the partial derivatives in the spatial variable. A technique to find the best value of the shape parameter is introduced. A high-resolution optimized hybrid block method is then used to solve the problem in the temporal variable. To validate the proposed method, several test problems are considered and the simulated results are compared with exact solutions and previous works. Moreover, a sensitivity analysis for parameter c is conducted, and the unconditional stability of the proposed algorithm has been validated.

In this paper, the stability of θ-methods for delay differential equations is studied based on the test equation $y^{\prime }\left(t\right)=-Ay\left(t\right)+By(t-\tau )$, where τ is a constant delay and A is a positive definite matrix. It is mainly considered the case where the matrices A and B are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices A and B are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

In recent years, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha <1$. An error analysis of the newly presented methods together with some numerical examples is provided at the end.

This paper deals with the numerical threshold stability of a nonlinear age-space structured heroin transmission model. A semi-discrete system is established by spatially domain discretization of the original nonlinear age-space structured model. A threshold value is proposed in stability analysis of the semi-discrete system and named as a numerical basic reproduction number. Besides the role it plays in numerical threshold stability analysis, the numerical basic reproduction number can preserve qualitative properties of the exact basic reproduction number and converge to the latter while stepsizes vanish. A fully discrete system is established via a time discretization of the semi-discrete system, in which an implicit-explicit technique is implemented to ensure the preservation of the biological meanings (such as positivity) without CFL restriction. Some numerical experiments are exhibited in the end to confirm the conclusions and explore the final state.