Low Dispersion Methods For Solving Wave Equations And Numerical Simulation

Numerical dispersion and computational efficiency are two key issues and difficulties for solving wave equations. In order to solve these problems, this dissertation employs the ideas of Padé approximation and nearly-analytic discrete and proposes Padé approximation method(PAM). We use Padé approximation method for the time discretization and obtain an implicit scheme, in which the time difference operator is a rational function. To avoid solving a large linear system with a block tridiagonal coefficient matrix, we propose an explicit method for this implicit algorithm, and then the implicit algorithm is replaced by explicit algorithm. For the spatial discretization, we adopt nearly-analytic discrete(NAD) operator, which uses linear combination of wave field displacements and their gradients to discretize higher order spatial derivatives. These discretization schemes contain much more wave field information which is beneficial for increasing the precision and imaging quality of seismic inversion and seismic migration. Meanwhile, this NAD operator has shorter operator radius and better compactness that correspond to the physical properties of the earth medium. With the combination of Padé approximation and nearly-analytic discrete method, the scheme has fourth-order accuracy in time and eighth-order accuracy in space. In addition, for the fourth and fifth order mixed partial differential operators, we apply the operator-splitting method that can effectively reduce the order of the differential operators appearing in the scheme and decrease the quantity of calculation.Firstly, we put forward the discrete formula using Padé approximation method for solving seismic wave equations. Afterwards, we analyze the stability condition, numerical relative error and dispersion relation for the one-dimensional and two-dimensional schemes. We also compare the computational efficiencies and numerical modeling results with the eighth-order Lax-Wendroff correction method and the eighth-order staggered-grid method. The results illustrate that the PAM has the properties of lower numerical dispersion and higher computational efficiency.Next, we apply Padé approximation method to solve three-dimensional seismic wave equations. For the spatial discretization, we use two different nearly-analytic discrete operators which are fourth-order and eighth-order in space, respectively. Comparisons of theoretical analysis and numerical experiments for three-dimensional wave equations with LWC and SG methods verify the validity of PAM for three-dimensional seismic wave equations and the properties of lower dispersion and higher computational efficiency. On the other hand, we generalize the so-called Runge-Kutta scheme to the three-dimensional case with eighth-order accuracy in space. Compared with the previous work, the eighth-order Runge-Kutta scheme also has higher order accuracy and lower numerical dispersion.In this dissertation, PAM is also generalized to solve two-phase Biot equation. The numerical scheme based on Padé approximation method has been derived and applied to solve low-frequency poroelastic wave equations in fluid-saturated porous media. We use the Padé approximation method for the time discretization and fourth-order NAD operator for the spatial discretization and then derive the PAM discrete formula. Numerical simulation results show that PAM can provide clear fast P, slow P and S waves in seismic wave fields. Compared with conventional numerical method such as LWC method, PAM has better capacity of suppressing numerical dispersion.