Linearly implicit methods for the nonlinear Klein–Gordon equation

Abstract

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.