Preconditioners for multilevel Toeplitz linear systems from steady-state and evolutionary advection-diffusion equations

Abstract

In this paper, we study preconditioners for multilevel Toeplitz linear systems arising from discretization of steady-state and evolutionary advection-diffusion equations, in which upwind scheme and central difference scheme are employed to discretize first-order and second-order terms, respectively. For the steady-state case, the preconditioner is constructed by replacing each of the discrete advection terms with a square root of the negative of discrete Laplacian matrix and the so constructed preconditioner is diagonalizable by a sine transform. Due to its diagonalizability, the preconditioner can be applied in a two-sided way. We prove that the GMRES solver for the preconditioned linear system has a linear convergence rate independent of discretization step-sizes. The sum of the time discretization and the steady-state preconditioner constitutes the evolutionary preconditioner. A fast implementation is proposed for the evolutionary preconditioner. Moreover, for the evolutionary case, we prove that the modulus of the eigenvalues of the preconditioned matrix is lower and upper bounded by positive constants independent of discretization step-sizes. We test the proposed preconditioners with several Krylov subspace solvers on some advection-dominated advection-diffusion problems and compare their performance with other preconditioners to show its efficiency.