# A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation

## TL;DR

This paper develops a preconditioning technique for solving all-at-once linear systems from nonlinear tempered fractional diffusion equations, improving computational efficiency and demonstrating convergence and effectiveness through numerical experiments.

## Contribution

It introduces a robust preconditioner for the nonlinear all-at-once system, enhancing the efficiency of Newton's method in solving these equations.

## Key findings

- The proposed schemes achieve first-order convergence in time and space.
- The preconditioner significantly accelerates the solution of Jacobian systems.
- Numerical results confirm the effectiveness of the preconditioning technique.

## 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.