A Preconditioning Technique for All-at-Once System from the Nonlinear Tempered Fractional Diffusion Equation

Abstract

An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are 1 in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may suitable for parallel computations.Newton's method, whose initial value is obtained by interpolating the solution of the linearized implicit scheme on the coarse space, is chosen to solve such the nonlinear all-at-once system. To accelerate the speed of solving the Jacobian equations appeared in Newton's method, a robust preconditioner is developed and analyzed. Numerical examples are reported to demonstrate the effectiveness of our proposed preconditioner.Meanwhile, they also imply that such the initial guess for Newton's method is more suitable.

On a Multigrid Method for Tempered Fractional Diffusion Equations

“…The inequality ( 42) is the well-known smoothing property [2]. We find that S h,ω ≤ 1, when ω satisfies (41). As A h,2 is symmetric, positive definite and diagonally dominant, we have…”

“…They also provided stability and convergence results for a second-order discretization of the tempered fractional diffusion equation. Zhao et al [41] designed the first-order fully implicit and semi-implicit schemes for the nonlinear tempered fractional diffusion equation with variable coefficients, where the stabilities and convergences of the two numerical schemes are proved under several assumptions. Then the PinT implementation of the fully implicit scheme is given and the resulting nonlinear system is solved by using the fast preconditioned iterative method.…”

All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations

“…In this regard, we mention the banded Toeplitz preconditioner proposed in [27] for solving non-linear space-FDEs, and the block structured preconditioner given in [3] for dealing with arbitrary dimensional space problems. In [10,21] a Strang-type preconditioner for solving FDEs by boundary value methods has been proposed.…”

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

“…To our knowledge, we can obtain numerical solutions of FPDEs globally in time by solving the all-atonce systems arising from FPDEs. This advantage attracts many researchers' attentions [18,[37][38][39][40][41][42][43]. Lu et al [37] proposed an approximate inversion (AI) method to solve the block lower triangular Toeplitz system with tri-diagonal blocks (BL3TB) from fractional sub-diffusion equations.…”