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We study the preconditioned iterative methods for the linear systems arising from the numerical solution of the multi‐dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed according to the symmetric and skew‐symmetric splitting of the Toeplitz factor in the resulting coefficient matrix. Theoretical analyses show that the upper bound of relative residual norm of the GMRES method when applied to the preconditioned linear system is mesh‐independent which implies the linear convergence. Numerical experiments are carried out to illustrate the correctness of the theoretical results and the effectiveness of the proposed preconditioning technique. [ABSTRACT FROM AUTHOR]

In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate. [ABSTRACT FROM AUTHOR]

For solving the initial‐boundary value problem of two‐dimensional wave equations with discrete and distributed time‐variable delays, in the present paper, we first construct a class of basic one‐parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one‐parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one‐parameter ADI method has the better advantage in computational efficiency than the basic one‐parameter methods. [ABSTRACT FROM AUTHOR]