The analysis techniques can be generalized to study the long-time behavior and asymptotic properties of the numerical solution beyond short-time error estimates by incorporating the following approaches:
Asymptotic Analysis: By extending the analysis to longer time scales, one can study the behavior of the numerical solution as it approaches equilibrium or steady-state solutions. Asymptotic techniques can be applied to understand the long-term dynamics.
Stability Analysis: Investigating the stability of the numerical scheme over extended time periods is crucial for ensuring the accuracy of the long-time behavior. Stability analysis techniques can be employed to study the behavior of the numerical solution over time.
Convergence in Time: Extending the error estimates to longer time intervals requires a detailed analysis of the convergence properties of the numerical scheme. By studying the convergence behavior over extended periods, one can infer the long-time properties of the solution.
Numerical Dissipation: Understanding the numerical dissipation effects over long time scales is essential for accurately capturing the dynamics of the Cahn-Hilliard equation. Analyzing the dissipation properties of the numerical scheme can provide insights into the long-term behavior of the solution.
By incorporating these approaches, the analysis can be generalized to study the long-time behavior and asymptotic properties of the numerical solution in a comprehensive manner.