  Among numerical methods for partial differential equations arising from
steepest descent dynamics of energy functionals (e.g., Allen-Cahn and
Cahn-Hilliard equations), the convex splitting method is well-known to maintain
unconditional energy stability for a large time step size. In this work, we
show how to use the convex splitting idea to find transition states, i.e.,
index-1 saddle points of the same energy functionals. Based on the iterative
minimization formulation (IMF) for saddle points (SIAM J. Numer. Anal., vol.
53, p1786, 2015), we introduce the convex splitting method to minimize the
auxiliary functional at each cycle of the IMF. We present a general principle
of constructing convex splitting forms for these auxiliary functionals and show
how to avoid solving nonlinear equations. The new numerical scheme based on the
convex splitting method allows for large time step sizes. The new methods are
tested for the one dimensional Ginzburg-Landau energy functional in the search
of the Allen-Cahn or Cahn-Hilliard types of transition states. We provide the
numerical results of transition states for the two dimensional
Landau-Brazovskii energy functional for diblock copolymers.
