Sound and soliton wave propagation in homogeneous and heterogeneous mediums with the new two-derivative implicit–explicit Runge–Kutta–Nyström method

Abstract

This paper derives a new family of implicit–explicit time-marching methods for PDEs with the second-order derivative in time. The present implicit method is based on the two-derivative Runge–Kutta–Nyström methods, which use a third-order time derivative of the solution. Although the current approach is implicit, it does not need to invert the coefficient matrix of the discretized system of equations. The stability properties are assessed using Fourier analysis for the model test problems by considering space–time discretizations together. The present methods are validated by comparing to some of the most widely used time-marching methods available in the literature. In addition, to assess the robustness and efficiency of the present methods, we have also performed numerical simulations of acoustic wave propagation in two- and three-layered heterogeneous media and sine-Gordon solitons for damped and undamped cases. Computed results match very well with the exact and numerical solutions noted in the literature.