Symplectic Stereomodelling Method For Wave Equations In Two-phase And Viscoelastic Media And Its Numerical Simulations

It has important theoretical and practical significance for oil/gas exploration tonumerically solve the seismic wave equations in fluid-saturated two-phase media andviscoelastic media and to simulate the wave fields. In this thesis, the symplectic methodfor solving wave equations in two-phase media and viscoelastic media are developedwithin the Birkoffian system.For equations describing waves propagating in two-phase media, this thesisconstructs the corresponding Birkhoffian system and the generalizedBirkhoffian system containing the gradients of the displacements andparticle-velocities. Based on the Birkhoffian system, the symplecticstereomodelling (SSM) method is suggested. This method uses the second-ordersymplectic geometric algorithm for time advancing and the fourth-orderstereomodelling operators (NAD and NAID operators) to approximate thehigh-order spatial derivatives. Based on such a structure, the SSM method cannot only preserve the symplectic geometry structure of the system, but alsosuppress effectively the numerical dispersion caused by discretizing the waveequations when coarse grids are used or models have large velocity contrastsbetween adjacent layers. Meanwhile, some theoretical problems (e.g. error,stability condition, numerical dispersion) for the SSM method are studied, andthe effectiveness and validity are investigated through numerical experiments.Besides, this thesis also presents the generalized Birkhoffian systems forthe P-wave and P-SV wave equations in two-dimensional viscoelastic media. Inorder to solve the Birkhoffian systems for the viscoelastic case, the SSM methodis developed. This method similarly employs the stereomodelling operators toapproximate the high-order spatial derivatives. Numerical simulations show thatthis method not only stays stable for the long time computations, but alsosuppress effectively the numerical dispersion.