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

An all-at-once system of nonlinear algebra equations arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, both the nonlinear and linearized implicit difference schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two numerical schemes are proved under several assumptions. Numerical examples show that the convergence orders of these two schemes are 1 in both time and space. Secondly, the nonlinear all-at-once system is derived from the nonlinear implicit scheme. 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 a 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 illustrate the effectiveness of our proposed preconditioner. Meanwhile, they also imply that our chosen initial guess for Newton’s method is feasible.