Weighted And Shifted Grunwald Difference Research Articles

Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation

In this article, some second-order time discrete schemes covering parameter 𝜃 combined with Galerkin finite element (FE) method are proposed and analyzed for looking for the numerical solution of nonlinear cable equation with time fractional derivative. At time tk−𝜃, some second-order 𝜃 schemes combined with weighted and shifted Grunwald difference (WSGD) approximation of fractional derivative are considered to approximate the time direction, and the Galerkin FE method is used to discretize the space direction. The stability of second-order 𝜃 schemes is derived and the second-order time convergence rate in L2-norm is proved. Finally, some numerical calculations are implemented to indicate the feasibility and effectiveness for our schemes.

WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation

In this paper, a new numerical approximation is discussed for the two-dimensional distributed-order time fractional reaction–diffusion equation. Combining with the idea of weighted and shifted Grunwald difference (WSGD) approximation (Tian et al. in Math Comput 84:1703–1727, 2015; Wang and Vong in J Comput Phys 277:1–15, 2014) in time, we establish orthogonal spline collocation (OSC) method in space. A detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order $$\mathscr {O}(\tau ^2+\Delta \alpha ^2+h^{r+1})$$ , where $$\tau , \Delta \alpha , h$$ and r are, respectively the time step size, step size in distributed-order variable, space step size, and polynomial degree of space. Interestingly, we prove that the proposed WSGD-OSC scheme converges with the second-order in time, where OSC schemes proposed previously (Fairweather et al. in J Sci Comput 65:1217–1239, 2015; Yang et al. in J Comput Phys 256:824–837, 2014) can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Some numerical results are also given to confirm our theoretical prediction.

PCG method with Strang’s circulant preconditioner for Hermitian positive definite linear system in Riesz space fractional advection–dispersion equations

In this paper, preconditioned conjugate gradient (PCG) method with Strang’s circulant preconditioner is investigated to solve the Hermitian positive definite linear systems, which is result from the Crank–Nicolson (C-N) finite difference scheme with the weighted and shifted Grunwald difference (WSGD) operators to discretize the Riesz space fractional advection–dispersion equation (RSFADE). We show that the spectrum of the preconditioned matrix is clustered around 1, and the singular values of the preconditioned matrix are uniformly bounded away from zero under a certain condition, respectively; hence the PCG method, when applied to solving the preconditioned system, converges superlinearly. Moreover, the complexity in each iteration of the PCG method is $$O(N\log N)$$ via using the fast Fourier transforms, where N is the matrix size. Numerical experiments are included to demonstrate the effectiveness of our approach.