An Energy Conserving Numerical Scheme for the Klein-Gordon Equation with Cubic Nonlinearity

In this paper, we develop a numerical scheme that conserves the discrete energy for solving the Klein-Gordon equation with cubic nonlinearity. We prove theoretically that our scheme conserves not just discrete energy, but also other energy-like discrete quantities. In addition, we prove the convergence and the stability of the scheme. Finally, we present some numerical simulations to demonstrate the performance of our energy-conserving scheme.

Automated summary

In this paper, a numerical scheme is developed to solve the Klein-Gordon equation with cubic nonlinearity while conserving the discrete energy. The theoretical proof demonstrates that the scheme also conserves other energy-like discrete quantities. Furthermore, the convergence and stability of the scheme are proven. Numerical simulations are presented to showcase the performance of the energy-conserving scheme.