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

A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method