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 + N^1?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.     

A second-order space–time accurate scheme for nonlinear diffusion equation with general capacity term

A fully implicit finite difference (FIFD) scheme with second-order space–time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays an important role in the analysis. It is established by choosing a new test function to bound the solution and its temporal and spatial difference quotients in suitable norms in the fixed point arguments, which is different from the traditional way. Based on these bounds, other fundamental properties of the scheme are rigorously analyzed consequently. It shows that the scheme is uniquely solvable, unconditionally stable, and convergent with second-order space–time accuracy in L^∞(L²) and L^∞(H¹) norms. The theoretical analysis adapts to both one- and multidimensional problems, and can be extended to schemes with first-order time accuracy. Numerical tests are provided to verify the theoretical results and highlight the high accuracy of the second-order space–time accurate scheme. The reasoning techniques can be extended to a broad family of discrete schemes for nonlinear problems with capacity terms.     

Finite difference spectral approximation for the time‐space fractional telegraph equation and its parameter estimation

The object of this paper is to present the numerical solution of the time‐space fractional telegraph equation. The proposed method is based on the finite difference scheme in temporal direction and Fourier spectral method in spatial direction. The fast Fourier transform (FFT) technique is applied to practical computation. The stability and convergence analysis are strictly proven, which shows that this method is stable and convergent with (2?α) order accuracy in time and spectral accuracy in space. Moreover, the Levenberg‐Marquardt (L‐M) iterative method is employed for the parameter estimation. Finally, some numerical examples are given to confirm the theoretical analysis.     

Compact finite difference scheme for the solution of time fractional advection-dispersion equation

In this paper, a compact finite difference method is proposed for the solution of time fractional advection-dispersion equation which appears extensively in fluid dynamics. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ ^2???α ), 0?<?α?<?1, and spatial derivatives are replaced with a fourth order compact finite difference scheme. We will prove the unconditional stability and solvability of proposed scheme. Also we show that the method is convergence with convergence order O(τ ^2???α ?+?h ⁴). Numerical examples confirm the theoretical results and high accuracy of proposed scheme.     

Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions

In this paper, the finite difference scheme is developed for the time-space fractional diffusion equation with Dirichlet and fractional boundary conditions. The time and space fractional derivatives are considered in the senses of Caputo and Riemann-Liouville, respectively. The stability and convergence of the proposed numerical scheme are strictly proved, and the convergence order is O(τ^2−α+h²). Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.     

A numerical technique based on B-spline for a class of time-fractional diffusion equation

This paper presents an efficient numerical technique for solving a class of time-fractional diffusion equation. The time-fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B-spline basis function is employed for discretization of space variable. The unconditional stability of the fully-discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)-th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.     

A numerical comparison of Chebyshev methods for solving fourth order semilinear initial boundary value problems

In solving semilinear initial boundary value problems with prescribed non-periodic boundary conditions using implicit-explicit and implicit time stepping schemes, both the function and derivatives of the function may need to be computed accurately at each time step. To determine the best Chebyshev collocation method to do this, the accuracy of the real space Chebyshev differentiation, spectral space preconditioned Chebyshev tau, real space Chebyshev integration and spectral space Chebyshev integration methods are compared in the L² and W^2,2 norms when solving linear fourth order boundary value problems; and in the L^∞([0,T];L²) and L^∞([0,T];W^2,2) norms when solving initial boundary value problems. We find that the best Chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space Chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to Fourier spectral discretization.     

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L²‐RH problems, we consider the construction of quarter plane solutions which are C¹ in time and C³ in space.     

Space‐time spectral collocation method for the one‐dimensional sine‐Gordon equation

In this article, we introduce a new space‐time spectral collocation method for solving the one‐dimensional sine‐Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second‐order system of ordinary differential equations (ODE). Our formulation has high‐order accurate in both space and time. Optimal a priori error bounds are derived in the L²‐norm for the semidiscrete formulation. Numerical experiments show that our formulation have exponential rates of convergence in both space and time. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 670–690, 2015     

Stability and error analysis for spectral Galerkin method for the Navier–Stokes equations with L2 initial data

In this article we consider the spectral Galerkin method with the implicit/explicit Euler scheme for the two‐dimensional Navier–Stokes equations with the L² initial data. Due to the poor smoothness of the solution on [0,1), we use the the spectral Galerkin method based on high‐dimensional spectral space H_M and small time step Δt² on this interval. While on [1,∞), we use the spectral Galerkin method based on low‐dimensional spectral space H_m(m = O(M^1/2)) and large time step Δt. For the spectral Galerkin method, we provide the standard H²‐stability and the L²‐error analysis. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Single and multi‐interval Legendre spectral methods in time for parabolic equations

In this article, we take the parabolic equation with Dirichlet boundary conditions as a model to present the Legendre spectral methods both in spatial and in time. Error analysis for the single/multi‐interval schemes in time is given. For the single interval spectral method in time, we obtain a better error estimate in L²‐norm. For the multi‐interval spectral method in time, we obtain the L²‐optimal error estimate in spatial. By choosing approximate trial and test functions, the methods result in algebraic systems with sparse forms. A parallel algorithm is constructed for the multi‐interval scheme in time. Numerical results show the efficiency of the methods. The methods are also applied to parabolic equations with Neumann boundary conditions, Robin boundary conditions and some nonlinear PDEs. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.     

Orthogonal spline collocation scheme for multiterm fractional convection‐diffusion equation with variable coefficients

The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in H^j (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

A Petrov‐Galerkin method with quadrature for semi‐linear second‐order hyperbolic problems

We propose and analyze a Crank–Nicolson quadrature Petrov–Galerkin (CNQPG) ‐spline method for solving semi‐linear second‐order hyperbolic initial‐boundary value problems. We prove second‐order convergence in time and optimal order H² norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order H^k, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives

In this study a new framework for solving three-dimensional (3D) time fractional diffusion equation with variable-order derivatives is presented. Firstly, a θ-weighted finite difference scheme with second-order accuracy is introduced to perform temporal discretization. Then a meshless generalized finite difference (GFD) scheme is employed for the solutions of remaining problems in the space domain. The proposed scheme is truly meshless and can be used to solve problems defined on an arbitrary domain in three dimensions. Preliminary numerical examples illustrate that the new method proposed here is accurate and efficient for time fractional diffusion equation in three dimensions, particularly when high accuracy is desired.     

Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation

In this paper, two novel linear-implicit and momentum-preserving Fourier pseudo-spectral schemes are proposed and analyzed for the regularized long-wave equation. The numerical methods are based on the blend of the Fourier pseudo-spectral method in space and the linear-implicit Crank–Nicolson method or the leap-frog scheme in time. The two fully discrete linear schemes are shown to possess the discrete momentum conservation law, and the linear systems resulting from the schemes are proved uniquely solvable. Due to the momentum conservative property of the proposed schemes, the Fourier pseudo-spectral solution is proved to be bounded in the discrete L^∞ norm. Then by using the standard energy method, both the linear-implicit Crank–Nicolson momentum-preserving scheme and the linear-implicit leap-frog momentum-preserving scheme are shown to have the accuracy of in the discrete L^∞ norm without any restrictions on the grid ratio, where N is the number of nodes and τ is the time step size. Numerical examples are carried out to verify the correction of the theory analysis and the efficiency of the proposed schemes.     

Single cell finite difference approximations of O(kh2 + h4) for ∂u/∂x for one space dimensional nonlinear parabolic equation

We report a new two‐level explicit finite difference method of O(kh² + h⁴) using three spatial grid points for the numerical solution of for the solution of one‐space dimensional nonlinear parabolic partial differential equation subject to appropriate initial and Dirichlet boundary conditions. The method is shown to be unconditionally stable when applied to a linear equation. The proposed method is applicable to the problems both in cartesian and polar coordinates. Numerical examples are provided to demonstrate the efficiency and accuracy of the method discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 408–415, 2000     

Error estimates of structure-preserving Fourier pseudospectral methods for the fractional Schrödinger equation

This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L² norm with the second-order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results.