Simple and accurate finite difference methods for the d-dimensional tempered fractional Laplacian and their applications.

In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional (d >=1) tempered fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems. Our finite difference methods have an accuracy of O(h^\epsilon), for u \in C^{0,\alpha+\epsilon} (R^d) if \alpha =1) with \epsilon > 0, suggesting the minimum consistency conditions. This accuracy can be improved to O(h^2), for u \in C^{2,\alpha+\epsilon}(R^d) if \alpha = 1). Numerical results confirm our analytical conclusions and provide further insights of our methods in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our methods only require the solution u \in C^{1,1}(R^d) for any 0 < \alpha < 2. Moreover, if the solution u \in C^{p, s}(R^d) for 0 <= p, s <=1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem. Compared to other existing methods, our methods have better accuracy and are simpler to implement. Since our methods yield a multilevel Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply them together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including nonlinear Schrodinger equation, Allen-Cahn equation, and Gray-Scott equations.