Convergence analysis and error estimates for the CSRK schemes to conserved gradient flows

We conduct a comprehensive convergence and error analysis for the second- and third-order unconditionally energy stable convex-splitting-Runge-Kutta (CSRK) methods for $H^{-1}$ gradient flows with typical forms of free energy. Through the energy structure inherent to gradient flows, we are able to derive uniform-in-time bounds of the numerical solution in the $H^1$, $H^2$, and $L^6$ norms. In turn, these functional bounds enable us to derive the associated estimates for the nonlinear error terms. Meanwhile, motivated by the fact that the diffusion coefficients are diagonally dominated in the CSRK numerical systems, the convergence results become available, based on a stage-by-stage analysis for the error evolutionary equations. The Cahn-Hilliard (CH) and phase field crystal (PFC) equations are two examples in the theoretical analysis. We also numerically compute some convergence results to validate the theorems proposed in this paper. This work deepens the theoretical foundation of CSRK methods and provides robust analytical tools for their application to conserved gradient flows.