Legendre Spectral Approximation Research Articles

High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates

<abstract><p>A coupled system of fractional order Gross-Pitaevskii equations is under consideration in which the time-fractional derivative is given in Caputo sense and the spatial fractional order derivative is of Riesz type. This kind of model may shed light on some time-evolution properties of the rotating two-component Bose¢ Einstein condensates. An unconditional convergent high-order scheme is proposed based on L2-$ 1_{\sigma} $ finite difference approximation in the time direction and Galerkin Legendre spectral approximation in the space direction. This combined scheme is designed in an easy algorithmic style. Based on ideas of discrete fractional Grönwall inequalities, we can prove the convergence theory of the scheme. Accordingly, a second order of convergence and a spectral convergence order in time and space, respectively, without any constraints on temporal meshes and the specified degree of Legendre polynomials $ N $. Some numerical experiments are proposed to support the theoretical results.</p></abstract>

On the existence and uniqueness of solutions to a nonlinear variable order time-fractional reaction–diffusion equation with delay

In this article, our purpose is to study the existence and uniqueness of a solution to a damped variable order fractional subdiffusion equation with time delay. Under weak assumptions on the data, we prove the uniqueness of a weak solution to the problem under consideration. The method of semi-discretization is extended to this kind of time fractional parabolic problem with delay in the case that the time delay parameter s>0 satisfies s⩽T, where T denotes the final time. As a consequence, two a priori estimates are predicted based on a discrete variational formulation of the problem. The existence of the problem’s weak solution on the time frame 0,⌊Ts⌋s is established by the aid of these derived a priori estimates. The paper is closed by introducing a fully discrete scheme based on Galerkin Legendre spectral approximation for the spatial operator and the backward Euler difference approximation for the temporal variable order operator. Accordingly, the accuracy and efficiency of the proposed scheme are justified by giving some numerical experiments for the sake of clearness.

An alternating direction implicit legendre spectral method for simulating a 2D multi-term time-fractional Oldroyd-B fluid type diffusion equation

In this paper, an alternating direction implicit legendre spectral (ADILS) method is developed for the two-dimensional (2D) multi-term time fractional Oldroyd-B fluid type diffusion equation. This method is provided based on legendre spectral approximation in space and a new difference discretization in time, and it can convert 2D computation to two 1D computations. Meanwhile, a very important lemma is proposed to prove the unconditional stability and convergence of this ADILS scheme. Especially, the corrected scheme is utilized to increase the convergence accuracy for the non-smooth solution. With three numerical experiments, the validity of the method is confirmed. The present method also can be extended to solve the generalized non-Newtonian fluid type equations, multi-term time fractional mixed diffusion and diffusion-wave equations.

Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation

The main aim of this paper is to construct an efficient Galerkin–Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrodinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre–Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of $$\min \{\kappa \theta ,2-\theta \}$$ order convergence in time and of spectral accuracy in space, where $$\theta $$ is the order of fractional derivative and $$\kappa $$ is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.

Fully discrete spectral method for solving a novel multi-term time-fractional mixed diffusion and diffusion-wave equation

A novel multi-term time-fractional mixed diffusion and diffusion-wave equation will be considered in this work. Different from the general multi-term time-fractional mixed diffusion and diffusion-wave equations, this new multi-term equation possesses a special time-fractional operator on the spatial derivative. We use a new discrete scheme to approximate the time-fractional derivative, which can improve the temporal accuracy. Then, a fully discrete spectral scheme is developed based on finite difference discretization in time and Legendre spectral approximation in space. Meanwhile, a very important lemma is proposed and proved, to obtain the unconditional stability and convergence of the fully discrete spectral scheme. Finally, four numerical experiments are presented to confirm our theoretical analysis. Both of our analysis and numerical test indicate that the fully discrete scheme is accurate and efficient in solving the generalized multi-term time-fractional mixed diffusion and diffusion-wave equation.

Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order gamma in (1,2). We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the L_{1} approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.

Spectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutions

A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative

Recently Caputo and Fabrizio introduced a new derivative with fractional order without singular kernel. The derivative can be used to describe the material heterogeneities and the fluctuations of different scales. In this article, we derived a new discretization of Caputo–Fabrizio derivative of order α (1 < α < 2) and applied it into the Cattaneo equation. A fully discrete scheme based on finite difference method in time and Legendre spectral approximation in space is proposed. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in H1 norm is O(τ2 + N1−m), where τ, N and m are the time‐step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Furthermore, the accuracy and applicability of the scheme are confirmed by numerical examples to support the theoretical results.

An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is O(N−r+τ2), where N,τ,r are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids.

Fully discrete spectral methods for solving time fractional nonlinear Sine–Gordon equation with smooth and non-smooth solutions

We consider the initial boundary value problem of the time fractional nonlinear Sine–Gordon equation and the fractional derivative is described in Caputo sense with the order α(1 < α < 2). Two fully discrete schemes are developed based on Legendre spectral approximation in space and finite difference discretization in time for smooth solutions and non-smooth solutions, respectively. Numerical stability and convergence are analysed. Numerical experiments for both the fully discrete schemes are presented to confirm our theoretical analysis.

A Fully Discrete Spectral Method for the Nonlinear Time Fractional Klein-Gordon Equation

The numerical approximation of the nonlinear time fractional Klein-Gordon equation in a bounded domain is considered. The time fractional derivative is described in the Caputo sense with the order $\gamma$ ($1 < \gamma < 2$). A fully discrete spectral scheme is proposed on the basis of finite difference discretization in time and Legendre spectral approximation in space. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in $H^1$ norm is $\mathrm{O}(\tau^{3-\gamma} + N^{1-m})$, where $\tau$, $N$ and $m$ are the time-step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Numerical examples are presented to support the theoretical results.

A Posteriori Error Estimation of Spectral and Spectral Element Methods for the Stokes/Darcy Coupled Problem

In this paper, we carry out an a posteriori error analysis of Legendre spectral approximations to the Stokes/Darcy coupled equations. The spectral approximations are based on a weak formulation of the coupled equations by using the Beavers-Joseph- Saffman interface condition. The main contribution of the paper consists of deriving a number of posteriori error indicators and their upper and lower bounds for the sin- gle domain case. An extension of the upper bounds to the multi-domain case in the spectral element framework is also given. AMS subject classifications: 35Q35, 76D03

Mixed Laguerre–Legendre Spectral Method for Incompressible Flow in an Infinite Strip

This paper concerns the mixed Laguerre–Legendre spectral approximation and its application to numerical simulation of incompressible flow in an infinite strip. Some approximation results in weighted Sobolev spaces are given. A Laguerre–Legendre spectral scheme for the stream function form of Navier–Stokes equations is constructed. The stability and the convergence of the proposed scheme are proved. The numerical experiments show the high accuracy of this method. The main techniques used in this paper are also applicable to other nonlinear partial differential equations in an infinite strip.

An Efficient Double Legendre Spectral Method for Parabolic and Elliptic Partial Differential Equations

We present a double Legendre spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomo-geneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. One numerical application of how to use these methods is described. Numerical results obtained compare favorably with those of the analytical solution. Accurate double Legendre spectral approximations for Poisson' and Helmholtz' equations are also noted.

The Fully Discrete Legendre Spectral Approximation of Two-Dimensional Unsteady Incompressible Fluid Flow in Stream Function Form

The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A fully discrete Legendre spectral scheme is proposed. By a series of a priori estimations and a compactness argument, it is proved that the numerical solution converges to the weak solution of the original problem. If the genuine solution is suitably smooth, then this approach provides higher accuracy. The numerical results show the advantages of this method. The techniques used in this paper are also applicable to other related problems with derivatives of high order in space.

Error Estimates for the Spectral Approximation of the Nonstationary Stokes Problem

This paper analyzes a discretization of the nonstationary Stokes problem with periodic-nonperiodic boundary conditions. Fourier–Legendre spectral approximations are considered with respect to the space variables, while the time discretization is based on the second-order, Crank–Nicolson scheme. Optimal convergence estimates are proved for the velocity field.