The Runge-Kutta Type Method For Solving Wave Equations And Its Simulation Of Seismic Wave Propagation

The forward modeling research of seismic wave propagation is one of the hotspot in the fields of Geophysics and petroleum exploration. In this thesis, we present a new numerical method, which are called Runge-Kutta method, to solve wave equations. We transform the second-order wave equations into a system of first-order partial differential equations and use not only the displacement and the particle-velocity of the grids, but also their gradients to approximate the high-order spatial derivatives, resulting that we get semi-discrete ordinary differential equations (ODEs). We then apply the Runge-Kutta method of order three or four to solve the semi-discrete ODEs and obtain our Runge-Kutta method for solving wave equations. In order to improve the Runge-Kutta method, we propose a weighted Runge-Kutta method. And we identically call the two methods"Runge-Kutta type method".In this thesis, we systemically investigate the Runge-Kutta type method including theoretical analyses and numerical computations. The main research contents are listed as follows. Firstly, the construction of Runge-Kutta methods to solve 1D, 2D, and 3D wave equations are respectively described, and the stability conditions of the 1D, 2D, and 3D Runge-Kutta methods are derived by Fourier analysis method. Secondly, the numerical error and dispersion of the 1D, 2D, and 3D Runge-Kutta methods are respectively analyzed, and the numerical accuracy and dispersion error of the Runge-Kutta method are also compared with those of the classic Lax-Wendroff correction method and the staggered-grid method. Thirdly, 2D and 3D numerical simulation results of acoustic and elastic wave propagation in different media are presented. Fourthly, the implementation steps and stability conditions of the weighted Runge-Kutta method with different weights are given. Finally, numerical simulation results of the seismic wave propagation are shown by the weighted Runge-Kutta method, meanwhile, the computation and storage efficiencies are investigated. The results of the theoretical analyses and numerical computations indicate that the Runge-Kutta method has good stability, high numerical accuracy, and small dispersion error, and can effectively suppress the numerical dispersion on coarse grids. And the weighted Runge-Kutta method can further save the storage and suppress the numerical dispersion, so the coarser spatial and temporal grids can be used to acquire faster computational speed and smaller storage. Therefore, the Runge-Kutta method and its weighted method have great potentiality of applying in Geophysics and petroleum exploration.