  We propose an easy-to-implement iterative method for resolving the implicit
(or semi-implicit) schemes arising in solving reaction-diffusion (RD) type
equations. We formulate the nonlinear time implicit scheme as a min-max saddle
point problem and then apply the primal-dual hybrid gradient (PDHG) method.
Suitable precondition matrices are applied to the PDHG method to accelerate the
convergence of algorithms under different circumstances. Furthermore, our
method is applicable to various discrete numerical schemes with high
flexibility. From various numerical examples tested in this paper, the proposed
method converges properly and can efficiently produce numerical solutions with
sufficient accuracy.
