On Preconditioners Based on HSS for the Space Fractional CNLS Equations

Abstract

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

Introduction to the Department of Cardiology in Nanjing First Hospital of Nanjing Medical University, China

“…Numerical results in [44] and [45] show that the PIHSS iteration method outperforms the HSS iteration method in terms of computing time, and the HSS-like iteration method has better behavior in terms of both iteration counts and computing time compared with the PIHSS iteration method. In [46], Ran et al came to a conclusion that the HSS-like preconditioner is more efficient than the HSS preconditioner and the AICD preconditioner when they are in conjunction with the Krylov subspace iteration methods. By taking the Toeplitz-plus-diagonal structure into account, Wang et al constructed an efficient variant of the PMHSS iteration method [47] which naturally leads to an efficient PMHSS preconditioner.…”

Algebraic multigrid block preconditioning for multi-group radiation diffusion equations

“…The objective is to approximately invert numerous individual scalar systems instead of the fully coupled systems. Preconditioners of this type had been proposed and analyzed in the literature, including the block diagonal preconditioner [11,22,53], block lower / upper triangular preconditioner [2,6,9], product (splitting) preconditioner [31,38,52] and constraint preconditioner [3,12,20]. Block preconditioners with multigrid components had proven very successful in a variety of applications, e.g., liquid crystal directors modeling [5], multiphase flow in porous media [7], Stokes problem [10], incompressible Navier-Stokes problem [13], second-order Agmon-Douglis-Nirenberg elliptic systems [21], magnetohydrodynamics model [23], Dirichlet biharmonic problem [29], electrical activity in the heart [36], Brinkman problem [37], all-speed melt pool flow physics [39] and fully coupled flow and geomechanics [40].…”

A Conservative Difference Scheme for Space Fractional Klein-Gordon-Schrödinger Equations with a High-Degree Yukawa Interaction

“…Refs. [4,7,8,18,22,24,25,32,35,54]. In particular, space fractional Schrödinger equations describe various physical phenomena [13,30,39,41].…”

“…In particular, space fractional Schrödinger equations describe various physical phenomena [13,30,39,41]. They are solved by conservative difference methods [33,43,44,[47][48][49], mass-conservative Fourier spectral methods [9], fourth-order methods [19,56], a collocation method [2], a conservative finite element method [23], and HSS-like iteration method [31,32]. The stability and convergence of these methods have been also investigated.…”