Nearly-analytic Symplectic PRK Method For Solving Wave Equations And Its Applications

Forward modeling method is an important basis of inversion problems. Currentlyin seismic exploration, one of the most widely applied forward methods is finitedifference. This dissertation proposes a new type of finite difference method for solvingseismic wave equations. At first, a new extended Hamiltonian system for waveequations is established by introducing the spatial gradients of the displacement andparticle velocity into the phase space of the Hamiltonian system. Subsequently, forsolving the extended Hamiltonian system, a new type of symplectic numerical methodwith low numerical dispersion is developed, which is called the nearly-analyticsymplectic partitioned Runge-Kutta (NSPRK) method. Afterwards, the theoreticalproperties of the NSPRK method are analyzed in detail, including the theoretical error,stability condition, and numerical dispersion relation. The results show that, comparedwith conventional symplectic method, although the NSPRK method has a relativelytighter stability condition, it can suppress numerical dispersion effectively.To verify the validity of NSPRK, numerical experiments for3D acoustic wave andelastic wave equations are made, which show that the numerical solutions generated byNSPRK are well coincident with the analytic solutions. Under the condition that visiblenumerical dispersions are eliminated, the computational efficiencies of NSPRK and fourconventional schemes are compared. The result shows that the NSPRK(2,4) is the mostefficient of them. For example, compared with the conventional symplectic methodSPRK(4,4), the computational speed of NSPRK(2,4) is increased by10.4times, whileits memory requirement and communication time between nodes in parallel are reducedto14.8%and7.6%, respectively.On the other hand, to resolve the artificial boundary problem in wave-fieldsimulation, the prevalent split-field perfectly matched layer (PML) absorbing boundarycondition is combined with the NSPRK in a new strategy. This strategy turns outeffective in reducing the reflected waves caused by the truncated boundaries. In addition,the convolutional PML for the second order seismic wave equation is derived. Thenumerical experiment shows that compared with the split-field PML, the convolutionalPML needs32%less memory storage, while its computational speed can be increasedby more than50times. Finally, NSPRK(2,4) and the two kinds of PML conditions are applied towave-field simulations in both2D and3D complex geological models. The resultsfurther prove the validity of NSPRK and its property of low numerical dispersion, andmoreover confirm the effectiveness of its combination with PML conditions. On theother hand, seismic wave propagation in different types of geological models withcomplex geological conditions is studied through these numerical simulations, and somenew recognization on wave propagation is obtained.