A Convergent Finite Difference Scheme for the Variational Heat Equation
This work provides a convergent numerical method for a nonlinear PDE arising in liquid crystal dynamics, addressing a known challenge in the field.
The authors developed a finite difference scheme for the variational heat equation and proved convergence of a subsequence of numerical solutions to a weak solution. Numerical examples demonstrated non-uniqueness of weak solutions and provided insight into selecting the physically relevant one.
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.