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

TL;DR

This paper introduces a high-precision spectral method-based numerical scheme for nonlinear hyperbolic evolution equations, specifically applied to the Klein-Gordon equation, achieving efficient computation with demonstrated accuracy improvements.

Contribution

The paper presents a novel spectral method-based numerical scheme with optimized computational cost for nonlinear hyperbolic equations, exemplified on the Klein-Gordon equation.

Findings

Numerical scheme achieves $O(N \, \log 2N)$ computational complexity.

Demonstrates the relationship between numerical precision and discretization size.

Provides benchmark results validating the scheme's accuracy.

Abstract

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.