Mass-Conserving Saddle Dynamics via Generalized Inner Product: Theory, Algorithms, and Applications

To reveal the effect of the inner product choice, we present a unified formulation of saddle dynamics for the functional F with a mass constraint under different inner products. We establish the equivalence between the index-k saddle points and the linearly stable steady states of the corresponding dynamics. Further, we present the dynamics with discrete H^{-1} and L^2 inner products and numerically verify the convergence orders of both dynamics. Finally, we apply the method to a phase field model with driving force under Neumann and periodic boundary conditions. The results uncover previously unreported saddle points and their connectivity, highlighting how the choice of inner product enriches the solution landscape in conservative systems.