Symplectic partitioned Runge–Kutta method based on the eighth-order nearly analytic discrete operator and its wavefield simulations
Zhang Chao-Yuan1,2, Ma Xiao2, Yang Lei2, and Song Guo-Jie3,4
1. College of Mathematics and Computer, Dali University, Dali 671003, China.
2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.
3. School of Science, Southwest Petroleum University, Chengdu 610500, China.
4. Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China.
Abstract：
We propose a symplectic partitioned Runge–Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic wave equation. Known as the eighth-order NSPRK method, this technique uses an eighth-order accurate nearly analytic discrete (NAD) operator to discretize high-order spatial differential operators and employs a second-order SPRK method to discretize temporal derivatives. The stability criteria and numerical dispersion relations of the eighth-order NSPRK method are given by a semi-analytical method and are tested by numerical experiments. We also show the differences of the numerical dispersions between the eighth-order NSPRK method and conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax–Wendroff correction (LWC) method and the eighth-order staggered-grid (SG) method. The result shows that the ability of the eighth-order NSPRK method to suppress the numerical dispersion is obviously superior to that of the conventional numerical methods. In the same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 times faster than the fourth-order SPRK, and the memory requirement is only approximately 47.17% of the fourth-order NSPRK method and 49.41 % of the fourth-order SPRK method, which indicates the highest computational efficiency. Modeling examples for the two-layer models such as the heterogeneous and Marmousi models show that the wavefields generated by the eighth-order NSPRK method are very clear with no visible numerical dispersion. These numerical experiments illustrate that the eighth-order NSPRK method can effectively suppress numerical dispersion when coarse grids are adopted. Therefore, this method can greatly decrease computer memory requirement and accelerate the forward modeling productivity. In general, the eighth-order NSPRK method has tremendous potential value for seismic exploration and seismology research.
ZHANG Chao-Yuan,MA Xiao,YANG Lei et al. Symplectic partitioned Runge–Kutta method based on the eighth-order nearly analytic discrete operator and its wavefield simulations[J]. APPLIED GEOPHYSICS, 2014, 11(1): 89-106.
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2014, Vol. 11 
