Stiff-cut leap-frog scheme for fractional Laplacian diffusion equations

Abstract

In this study, we investigate the numerical discretization of two-dimensional fractional Laplacian diffusion equations. After discretizing the space derivative, we split the discrete fractional Laplacian into two parts. One is the τ preconditioner multiplying a constant, solved using the leap-frog scheme, while the other part and the remaining nonlinear term are treated with an explicit scheme. This results in what is termed the stiff-cut leap-frog scheme, which is more efficient than the traditional leap-frog scheme. The main advantage of the proposed scheme is that the coefficient matrices of the generated linear systems belong to τ algebra, which can be diagonalized by fast discrete sine transform with the complexity of O(NlogN), where N is the freedom of unknowns. Hence, no iterative method is required at each time integration. Moreover, we theoretically prove that the proposed scheme is unconditionally stable and converges in second order. Ultimately, numerical experiments are conducted to demonstrate the precision and effectiveness of the stiff-cut leap-frog scheme.