Numerical Algorithms最新文献

In this paper, we propose a novel fully implicit numerical scheme that satisfies both nonlinear energy stability and maximum principle for the space fractional Allen-Cahn equation. Especially, the fully implicit second-order scheme in time has never been proved to preserve the maximum principle before. For the resulting nonlinear scheme, we also propose a nonlinear iterative algorithm, which is uniquely solvable, convergent, and can preserve discrete maximum principle in each iterative step. Then we provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are presented to verify the theoretical results.

The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-Sahalia type model arising in mathematical finance and evolving in a positive domain ((0, infty )), which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term (alpha _{-1} x^{-1}) and a corrective mapping (Phi _h) in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size (h>0)) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.

This article provides an effective numerical approach using the fractional integral operational matrix method for a fractional Legendre wavelet to deal with multi-dimensional fractional optimal control problems. We proposed operational matrices and implemented them to simplify multi-dimensional fractional optimal control problems into a set of equations, utilizing well-known formulas such as the Caputo-Fabrizio operator with a non-singular kernel defined for calculating fractional derivatives and integrals of fractional Legendre wavelets. Finally, the Lagrange multiplier technique is applied, and we get the state and control functions. The convergence analysis and error bounds of the proposed scheme are established. To check the veracity of the presented method, we tested numerical examples using the fractional Legendre wavelet method and obtained the cost function value based on identifying state and control functions.

This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time ((t=0)) is encountered in the considered problem. To overcome this, we consider the high-order L2-1(_sigma ) formula on a suitably designed non-uniform fitted mesh, to discretize the time-fractional derivative. Further, a high-order two-dimensional compact operator is developed to approximate the spatial variables. Moreover, an alternating direction implicit (ADI) approach is designed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, is conducted comprehensively. More precisely, it is shown that method is convergent of order (mathcal Oleft( {N_t^{-min {3-alpha ,theta alpha ,1+alpha ,2}}}+h_x^4+h_y^4right) ), where (alpha in (0,1)) represents the order of the fractional derivative, (theta ) is a parameter which is utilized in the construction of the fitted mesh, (N_t) is the temporal discretization parameter, and (h_x) and (h_y) represent the spatial mesh widths. The numerical outcomes for three test problems, each featuring the nonsmooth solution, verified the theoretical findings. Further, the proposed method on fitted meshes exhibits superior numerical accuracy in comparison to the existing methods.

In this paper, we construct numerical methods of fourth, sixth and eighth orders convergence for solving fully fourth order nonlinear differential equation with the Dirichlet boundary conditions. The methods are based on the use of the trapezoidal quadrature formula with corrections for computing integrals at each iteration of the continuous iterative method for finding the solutions of the BVP. We get the error estimates for the actually obtained numerical solutions of the problem. Many numerical examples confirm the theoretical conclusions and show the efficiency of the proposed methods in comparison with some existing methods.

Based on the diagonal and off-diagonal splitting (DOS) iteration scheme (Dehghan et al. Filomat 31(5), 1441–1452 2017), we offer an iteration procedure called Uzawa-DOS to solve a class of saddle-point problems (SPPs). Each iteration of this iterative method involves two subsystems with diagonal and lower triangular matrices. Due to the simple structure of involved coefficient matrices, two linear subsystems are solvable exactly, which is a notable precedence of the Uzawa-DOS method and can make it inexpensive to execute. Theoretical analysis verifies convergence of the proposed method under appropriate conditions. The suggested method is validated by numerical experiments.

In this paper, by regarding the two-subspace Kaczmarz method as an alternated inertial randomized Kaczmarz algorithm we present a better convergence rate estimate under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertial randomized Kaczmarz algorithm which is proved to have a faster convergence rate. Numerical experiments support the theory results and illustrate that the multi-inertial randomized Kaczmarz algorithm significantly outperform the two-subspace Kaczmarz method in solving coherent linear systems.

The Caputo fractional derivative of a real continuous function g distinguishes from the other fractional derivative methods with the demand for the existence of its first order derivative (g'). This attribute leads to the investigation of Caputo fractional derivative of (alpha )-fractal splines rather than just a continuous non-differentiable (alpha )-fractal function. A bounded linear operator corresponding to the Caputo fractional derivative of fractal version is reported. In addition, a new family of fractal perturbations is proposed in association with the fractional derivative. Thereafter, a numerical approach is used to determine the exact Caputo fractional derivative of fractal functions in terms of Legendre polynomials.

The Galerkin method is proposed for initial value problem of auto-convolution Volterra integro-differential equation (AVIDE). The solvability of the Galerkin method is discussed, and the uniform boundedness of the numerical solution is provided by defining a discrete weighted exponential norm. In particular, it is proved that the quadrature Galerkin method obtained from the Galerkin method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the continuous piecewise polynomial collocation method. For the Galerkin approximated solution in continuous piecewise polynomial space of degree (varvec{m}), at first, the (varvec{m}) global convergence order is obtained. By defining a projection operator, the convergence is improved, and the optimal (varvec{m+1}) global convergence order is gained, as well as (varvec{2m}) local convergence order at mesh points. Furthermore, all the above analysis for uniform mesh can be extended to typical quasi-uniform meshes. Some numerical experiments are given to illustrate the theoretical results.

This paper aims to propose a new splitting mixed finite element method (MFE) for solving viscoelastic wave equations and give convergence analysis. First, by introducing two new variables (q=u_t) and (varvec{sigma }=A(x)nabla u+B(x)nabla u_t), a new system of first-order differential-integral equations is derived from the original second-order viscoelastic wave equation. Then, the semi-discrete and fully-discrete splitting MFE schemes are proposed by using the MFE spaces and the second-order time discetization. By the two schemes the approximate solutions for the unknowns u, (u_t) and (sigma ) are obtained simultaneously. It is proved that the semi-discrete and fully-discrete schemes have the optimal error estimates in (L^2)-norm. Meanwhile, it is proved that the fully-discrete SMFE scheme based on the Raviart-Thomas mixed finite element spaces and the uniform rectangular mesh partitions is super convergent. Finally, numerical experiments to compute the (L^2) errors for approximating u, q and (varvec{sigma }) and their convergence rates are presented, and the theoretical analysis on error estimates and convergence is then confirmed.