Pin Lyu

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.

A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

In this paper, a compact finite difference scheme with global convergence order

A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations

O\big (\tau ^{2-\alpha }+h^4\big )

Second-order BDF time approximation for Riesz space-fractional diffusion equations

O(τ2-α+h4) is derived for fractional sub-diffusion equations with the spatially variable coefficient subject to Neumann boundary conditions. The difficulty caused by the variable coefficient and the Neumann boundary conditions is overcome by subtle decomposition of the coefficient matrices. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. The theoretical results are supported by numerical experiments.

A linearized second-order finite difference scheme for time fractional generalized BBM equation

Two fully discrete methods are investigated for simulating the distributed-order sub-diffusion equation in Caputo’s form. The fractional Caputo derivative is approximated by the Caputo’s BDF1 (called L1 early) and BDF2 (or L1-2 when it was first introduced) approximations, which are constructed by piecewise linear and quadratic interpolating polynomials, respectively. It is shown that the first scheme, using the BDF1 formula, possesses the discrete minimum-maximum principle and nonnegativity preservation property such that it is stable and convergent in the maximum norm. The method using the BDF2 formula is shown to be stable and convergent in the discrete H1 norm by using the discrete energy method. For problems of distributed order within a certain region, the method is also proven to preserve the discrete maximum principle and nonnegativity property. Extensive numerical experiments are provided to show the effectiveness of numerical schemes, and to examine the initial singularity of the solution. The applicability of our numerical algorithms to a problem with solution which lacks the smoothness near the initial time is examined by employing a class of power-type nonuniform meshes.

On numerical contour integral method for fractional diffusion equations with variable coefficients

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.

A study on a second order finite difference scheme for fractional advection-diffusion equations

We study linearized finite difference scheme for a time-fractional Burgers-type equation in this paper. A linearized scheme with second-order accuracy in time and space is proposed. The advantage of the scheme is that iterative method is not required for finding the approximated solutions. Nonlinearity involving derivatives causes difficulties in analysis. By refined estimates of our previous study, we show that the scheme unconditionally converges with second-order in maximum-norm. The theoretical results are justified by numerical tests.

A linearized and second-order unconditionally convergent scheme for coupled time fractional Klein-Gordon-Schrödinger equation

ABSTRACT Second-order backward difference formula (BDF2) is considered for time approximation of Riesz space-fractional diffusion equations. The Riesz space derivative is approximated by the second-order fractional centre difference formula. To improve the computational efficiency, an alternating directional implicit scheme is also proposed for solving two-dimensional space-fractional diffusion problems. Numerical experiments are provided to verify our theory and to show the effectiveness of numerical algorithms.