Numerical scheme based on the spectral method for calculating nonlinear hyperbolic evolution equations

High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with the order of total calculation cost $O(N log 2N)$ is proposed. As benchmark results, the relation between the numerical precision and the discretization unit size are demonstrated.