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

TL;DR

This paper introduces a flexible and efficient primal-dual hybrid gradient algorithm for solving reaction-diffusion equations formulated as min-max saddle point problems, demonstrating good convergence and accuracy in numerical tests.

Contribution

The paper presents a novel iterative primal-dual hybrid gradient method tailored for implicit schemes in reaction-diffusion equations, with adaptable preconditioning for faster convergence.

Findings

Method converges properly in numerical experiments

Efficiently produces accurate solutions

Applicable to various numerical schemes

Abstract

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.