Effective Methods for Solving Band SLEs after Parabolic Nonlinear PDEs

TL;DR

This paper introduces effective numerical methods for solving banded systems of linear equations arising from finite difference discretizations of nonlinear heat transfer PDEs in multilayer domains, focusing on challenging non-diagonally dominant matrices.

Contribution

It develops two novel methods—diagonal dominantization and symbolic algorithms—for efficiently solving complex band SLEs from nonlinear PDE discretizations.

Findings

The methods successfully solve pentadiagonal systems with non-positive definite matrices.

Numerical experiments demonstrate improved stability and efficiency.

The approaches are applicable to multilayer heat transfer models with variable properties.

Abstract

A class of models of heat transfer processes in a multilayer domain is considered. The governing equation is a nonlinear heat-transfer equation with different temperature-dependent densities and thermal coefficients in each layer. Homogeneous Neumann boundary conditions and ideal contact ones are applied. A finite difference scheme on a special uneven mesh with a second-order approximation in the case of a piecewise constant spatial step is built. This discretization leads to a pentadiagonal system of linear equations (SLEs) with a matrix which is neither diagonally dominant, nor positive definite. Two different methods for solving such a SLE are developed -- diagonal dominantization and symbolic algorithms.