A nearly analytic symplectic partitioned Runge-Kutta method based on a locally one-dimensional technique for solving two-dimensional acoustic wave equations

摘要

In this paper, we develop a new nearly analytic symplectic partitioned Runge-Kutta method based on locally one-dimensional technique for numerically solving two-dimensional acoustic wave equations. We first split two-dimensional acoustic wave equation into the local one-dimensional equations and transform each of the split equations into a Hamiltonian system. Then, we use both a nearly analytic discrete operator and a central difference operator to approximate the high-order spatial differential operators, which implies the symmetry of the discretized spatial differential operators, and we employ the partitioned second-order symplectic Runge-Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations, which results in fully discretized scheme is symplectic unlike conventional nearly analytic symplectic partitioned Runge-Kutta methods. Theoretical analyses show that the nearly analytic symplectic partitioned Runge-Kutta method based on locally one-dimensional technique exhibits great higher stability limits and less numerical dispersion than the nearly analytic symplectic partitioned Runge-Kutta method. Numerical experiments are conducted to verify advantages of the nearly analytic symplectic partitioned Runge-Kutta method based on locally one-dimensional technique, such as their computational efficiency, stability, numerical dispersion and long-term calculation capability.