A nearly analytic symplectically partitioned Runge–Kutta method for 2-D seismic wave equations

In this paper, we develop a new nearly analytic symplectically partitioned Runge–Kutta (NSPRK) method for numerically solving elastic wave equations. In this method, we first transform the elastic wave equations into a Hamiltonian system, and use the nearly analytic discrete operator to approximate the high-order spatial differential operators, and then we employ the partitioned second-order symplectic Runge–Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations (ODEs). We investigate in great detail on the properties of the NSPRK method that includes the stability condition for the P–SV wave in a 2-D homogeneous isotropic medium, the computational efficiency, and the numerical dispersion relation for the 2-D acoustic case. Meanwhile, we apply the NSPRK to simulate the elastic wave propagating in several multilayer models with both strong velocity contrasts and fluctuating interfaces. Both theoretical analysis and numerical results show that the NSPRK can effectively suppress the numerical dispersion resulted from the discretization of the wave equations, and more importantly, it preserves the symplecticity structure for long-time computation. In addition, numerical experiments demonstrate that the NSPRK is effective to combine the split perfectly matched layer boundary conditions to take care of the reflections from the artificial boundaries.