A Time-Evolution Algorithm for Partial Differential Equations Derived From the Steady-State Solution of the Brusselator Model

For the two-dimensional Brusselator model, this paper obtains the steady-state solution of Turing patterns by using a time-evolution algorithm. Based on the research of the Brusselator model, we propose a new time-evolution algorithm for the boundary value problem of partial differential equations (PDEs). By introducing a virtual time parameter, the original boundary value problem is transformed into an initial value problem. Combining the finite difference method and the implicit iterative algorithm, the system evolution is simulated to approximate the steady-state solution, that is, the solution of the boundary value problem of the original equation set. The research shows that compared with the traditional Jacobi iterative method and Gauss-Seidel iterative method, the time-evolution algorithm exhibits stronger stability and convergence in complex non-linear problems. Further, this method is extended to the solution of Poisson's equation and Helmholtz equation. The results show that although the accuracy and efficiency of the time-evolution algorithm in general linear problems are slightly inferior to those of the Gauss-Seidel iterative method, in specific scenarios such as the Helmholtz equation with large wave numbers, its convergence speed and error accuracy are better. The time-evolution algorithm provides a new solution idea for the boundary value problem of non-linear PDEs. In the future, its performance can be further improved by combining high-order difference schemes and parallel computing strategies.