  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.
