A unified convergence analysis framework of the energy-stable ETDRK3 schemes for the No-slope-selection thin film model

TL;DR

This paper develops a comprehensive convergence analysis framework for energy-stable third-order exponential time differencing schemes applied to the No-slope-selection thin film model, combining Fourier spectral methods and eigenvalue analysis.

Contribution

It introduces a unified approach for analyzing the convergence of ETDRK3 schemes for the NSS model, addressing nonlinear complexities with eigenvalue bounds.

Findings

Derived optimal convergence rates and error estimates.

Established a rigorous Fourier eigenvalue analysis framework.

Resolved nonlinear term challenges through eigenvalue bounds.

Abstract

This paper establishes a unified framework for the space-time convergence analysis of the energy-stable third-order accurate exponential time differencing Runge-Kutta schemes. By employing Fourier pseudo-spectral discretization in space and the inner product technique, we derive a rigorous Fourier eigenvalue analysis, which provides a detailed optimal convergence rate and error estimate. The primary challenge is addressing the complex nonlinear terms in the NSS equation. Fortunately, this challenge could be resolved through careful eigenvalue bound estimates for various operators.