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.

研究了由变系数的非线性回火分数扩散方程组成的全一次非线性代数方程组。首先，提出了非线性和线性化的隐式差分格式，以逼近具有连续/不连续系数的非线性方程。在几个假设下证明了这两种数值格式的稳定性和收敛性。数值算例表明，这两种方案的收敛阶在时间和空间上均为1。其次，从非线性隐式方案中导出非线性一次系统。牛顿法的初值是通过在粗糙空间上内插线性化隐式方案的解而获得的，因此它被用来求解这种非线性的一次一次系统。为了加快牛顿方法中出现的Jacobian方程的求解速度，开发并分析了一种健壮的预处理器。报告了数值示例，以说明我们提出的预处理器的有效性。同时，它们也暗示我们对牛顿法选择的初始猜测是可行的。