A modified symplectic scheme for seismic wave modeling

Abstract

Symplectic integrators are well known for their excellent performance in solving partial differential equation of dynamical systems because they are capable of preserving some conservative properties of dynamic equations. However, there are not enough high-order, for example third-order symplectic schemes, which are suitable for seismic wave equations. Here, we propose a strategy to construct a symplectic scheme that is based on a so-called high-order operator modification method. We first employ a conventional two-stage Runge–Kutta–Nyström (RKN) method to solve the ordinary differential equations, which are derived from the spatial discretization of the seismic wave equations. We then add a high-order term to the RKN method. Finally, we obtain a new third-order symplectic scheme with all positive symplectic coefficients, and it is defined based on the order condition, the symplectic condition, the stability condition and the dispersion relation. It is worth noting that the new scheme is independent of the spatial discretization type used, and we simply apply some finite difference operators to approximate the spatial derivatives of the isotropic elastic equations for a straightforward discussion. For the theoretical analysis, we obtain the semi-analytic stability conditions of our scheme with various orders of spatial approximation. The stability and dispersion properties of our scheme are also compared with conventional schemes to illustrate the favorable numerical behaviors of our scheme in terms of precision, stability and dispersion characteristics. Finally, three numerical experiments are employed to further demonstrate the validity of our method. The modified strategy that is proposed in this paper can be used to construct other explicit symplectic schemes.