The analysis techniques proposed in the context can be extended to other high-order numerical schemes for the Cahn-Hilliard equation or similar nonlinear parabolic equations by following a similar framework.
Spatial Discretization: The use of high-order finite difference schemes, such as the fourth-order scheme discussed, can be applied to other nonlinear parabolic equations. These schemes provide improved accuracy and can capture finer details in the solution.
Temporal Discretization: Techniques like the modified BDF2 temporal discretization can be adapted to other high-order schemes. By carefully analyzing the stability and error properties of the temporal discretization, one can ensure accurate and stable numerical solutions.
Error Estimates: The approach of deriving refined error estimates, as seen in the context, can be applied to other high-order schemes. By carefully analyzing the error terms and applying suitable inequalities, one can obtain improved convergence constants.
Spectral Analysis: Utilizing spectrum analysis for linearized operators can help in understanding the stability and convergence properties of the numerical schemes. This technique can be extended to other high-order schemes for a thorough analysis.
Overall, by applying similar techniques and methodologies to other high-order numerical schemes, one can enhance the accuracy, stability, and convergence properties of the numerical solutions for nonlinear parabolic equations.