Arbitrary-order symplectic time integrator for the acoustic wave equation using the pseudo-spectral method

학위논문 (박사)-- 서울대학교 대학원 : 협동과정 계산과학전공, 2017. 2. 신창수.A Hamiltonian system is symplectic. To simulate a Hamiltonian system, symplectic time integrators are generally appliedotherwise, the energy or the generalized energy is not conserved in the volume of interest. In this study, the symplectic nature of the acoustic wave system is proven. Then, a symplectic scheme that can be extended arbitrarily in temporal dimensions is suggested. The method is based on the Lax-Wendroff expansion of the time differentiation of acoustic wave variables, such as pressure and velocity, existing on the staggered time axis, i.e., one is on the integer grid, and the other is defined on the half integer of the time step. The series can be reduced to the pseudo-differential operator, which enables the application of other approximation techniques, such as the Jacobi-Anger expansion. By virtue of considering the property of the nature of the acoustic wave phenomena, the scheme is more stable and accurate than methods that do not consider symplecticity. Moreover, the phase error per time step can be kept sufficiently small to conduct simulation over long periods of time. According to the analysis of the scheme, the larger the time strides are, the more efficient the simulation is in terms of computing power when a sufficient number of multiplications of the map are accumulated. The effectiveness and accuracy are verified through simulation results using a homogeneous model in which the computed wavefield is equivalent to the analytic solution. The numerical results of the wavefield in the heterogeneous model also yield equivalent results irrespective of the time step lengths. The scheme can be applied to the source problemshowever, the time step is confined to describing the entire frequency component of the wavelet.1. Introduction 1 1.1. Background 1 1.2. Overview 8 1.3. Outline 10 2. Theory 11 2.1. Acoustic wave equation 11 2.2. Symplecticity and symplectic time integrator 18 2.2.1. Symplecticity of the transformation map 18 2.2.2. Symplectic time integrator 21 2.3. Arbitrary-order symplectic time integrator 26 3. Analysis 31 3.1. Stability analysis 32 3.2. Dispersion analysis 37 3.3. Phase analysis 45 3.4. Spectral accuracy and compromise 56 3.5. Source wavelet issue 67 4. Numerical Examples 70 4.1. Initial value problems 71 4.1.1. Homogeneous model 71 4.1.2. Synthetic heterogeneous model: Marmousi-2 72 4.2. Source problems 90 4.2.1. Homogeneous model 90 4.2.2. Synthetic heterogeneous model: Marmousi-2 91 4.3. Discussion on factors debasing the accuracy 107 5. Conclusions 112 References 115 Appendix A. Additional formulations 121 A1. Absorbing boundary conditions 121 A2. Analytic solution 126 Appendix B. Matlab codes 128 B1. Arbitrary-order symplectic time operator 128 B2. Analytic solution 131 초록 133Docto