A class of one-stage two-derivative diagonally implicit Runge-Kutta methods for sine-Gordon equation

Abstract

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation with important applications in physics, biology, and engineering. Its dispersive and oscillatory nature makes accurate numerical simulation challenging, especially when using traditional time integration methods. This research develops and analyzes a class of phase-fitted one-stage two-derivative diagonally implicit Runge-Kutta (TDDIRK) methods for solving the sine-Gordon equation efficiently. In this paper, we use the method of lines to obtain a semi-discrete continuous system. A class of one-stage two-derivative DIRK methods are derived by optimizing their coefficients to reduce phase-lag and amplification errors. The stability and phase analyses are performed to examine their properties. Numerical experiments on kink and antikink soliton problems show that the proposed phase-fitted methods achieve higher accuracy and improved efficiency compared to non-optimized schemes. The results demonstrate that these methods are effective for solving oscillatory nonlinear partial differential equations like the sine-Gordon equation.