Unconditionally energy-stable discontinuous Galerkin method for the dynamics model of protein folding

In this paper, we present the coupled nonlinear Schrödinger equations to describe the conformational dynamics of protein secondary structure. We first construct a structure-preserving discrete scheme that ensures both mass conservation and energy stability. The proposed scheme is employed by combining the discontinuous Galerkin (DG) method for spatial discretization, Crank-Nicolson (C-N) approximation for temporal discretization, a second-order convex-concave splitting for the double-well potential and adding additional stabilization term. Moreover, by using the Brouwer fixed point theorem and the Gagliardo-Nirenberg inequality, we rigorously prove the unique solvability and convergence with second-order accuracy in both time and space without the grid ratio condition. Finally, numerical experiments are carried out to demonstrate the convergence rate, mass conservation, energy stability and performance of the developed scheme.