Structure-preserving scheme for 1D KWC system

TL;DR

This paper develops a numerical scheme for the 1D KWC system that preserves key structural properties like range and energy dissipation, ensuring accurate and stable simulations of grain boundary motion.

Contribution

It introduces a novel structure-preserving numerical scheme for the 1D KWC system, with proven properties and convergence analysis.

Findings

The scheme preserves the range of the solution.

The scheme guarantees energy dissipation.

Convergence conditions and error estimates are established.

Abstract

In this paper, we consider a system of one-dimensional parabolic PDEs, known as the KWC system, as a phase-field model for grain boundary motion. A key feature of this system is that the equation for the crystalline orientation angle is described as a quasilinear diffusion equation with variable mobility. The goal of this paper is to establish a structure-preserving numerical scheme for the system, focusing on two main structural properties: $\sharp\,1)$ range preservation; and $\sharp\,2)$ energy dissipation. Under suitable assumptions, we construct a structure-preserving numerical scheme and address the following in the main theorems: (O) verification of the structural properties; (I) clarification of the convergence conditions; and (II) error estimate for the scheme.