A Convergent Finite Difference Scheme for the Variational Heat Equation

TL;DR

This paper introduces a finite difference scheme for a transformed variational heat equation, proves convergence to a weak solution, and provides numerical examples illustrating non-uniqueness and physical relevance of solutions.

Contribution

The paper develops a convergent finite difference scheme for a nonlinear, non-divergence form variational heat equation and analyzes solution non-uniqueness.

Findings

Numerical solutions converge to a weak solution.

Weak solutions are not unique.

Numerical experiments provide insight into physically relevant solutions.

Abstract

The variational heat equation is a nonlinear, parabolic equation not in divergence form that arises as a model for the dynamics of the director field in a nematic liquid crystal. We present a finite difference scheme for a transformed, possibly degenerate version of this equation and prove that a subsequence of the numerical solutions converges to a weak solution. This result is supplemented by numerical examples that show that weak solutions are not unique and give some intuition about how to obtain the physically relevant solution.