Linearly implicit methods for the nonlinear Klein-Gordon equation

We present energy-preserving linearly implicit integrators for the nonlinear Klein-Gordon equation, based on the polarization of the polynomial functions. They are symmetric, second- order accurate in time and space, and unconditionally stable. Instead of solving a nonlinear algebraic equation at every time step, the linearly implicit integrators only require solving a linear system, which reduces the computational cost. We propose three types of linearly implicit integrators for the nonlinear Klein-Gordon equation, that preserve the modified, polarized invariants, ensuring the stability of the solutions in long-time integration. Numerical results confirm the theoretical convergence orders and preservation of the Hamiltonians that guarantee the stability of the solutions in long-time simulation.