Banded preconditioners for Riesz space fractional diffusion equations
In this paper, numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations are considered. Crank-Nicolson technique is applied to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. They transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. They prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners. In my opinion, this work is interesting and makes some progress in fast algorithm for solving fractional partial differential equation.
Riesz space fractional diffusion equation
scientific article; zbMATH DE number 7352176
scientific article