# Numerical solution of nonlinear parabolic systems by block monotone iterations

## TL;DR

This paper develops and analyzes block monotone iterative methods, based on Jacobi and Gauss-Seidel schemes, for efficiently solving coupled nonlinear parabolic systems with monotone reaction functions.

## Contribution

It introduces a block monotone iterative framework for nonlinear parabolic systems, proving convergence and solution existence using upper and lower solution sequences.

## Key findings

- Block Gauss-Seidel converges at least as fast as block Jacobi.
- Monotone sequences guarantee existence of solutions for quasi-monotone systems.
- The methods effectively handle coupled nonlinear parabolic problems.

## Abstract

This paper deals with investigating numerical methods for solving coupled system of nonlinear parabolic problems. We utilize block monotone iterative methods based on Jacobi and Gauss--Seidel methods to solve difference schemes which approximate the coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing or nonincreasing. In the view of upper and lower solutions method, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence of solutions to problems with quasi-monotone nondecreasing and nonincreasing reaction functions. Construction of initial upper and lower solutions is presented. The sequences of solutions generated by the block Gauss--Seidel method converge not slower than by the block Jacobi method.