Research On The Application Of Modified Symplectic Algorithm In The Calculation Of Wave Equation

The development of efficient numerical methods for solving wave equations has important practical significance,which affects all aspects of production and life.Solving partial differential equations with a computer involves both temporal discretization and spatial discretization.At present,the spatial discretizations have been explored extensively.Comparison of spatial discretization,temporal discretization has been payed less attention.Numerical schemes for temporal derivatives also introduce numerical artifacts,which are called time-dispersion error.It has great research value to study the time-discrete format matching with spatial discretization.Here,we focus on the temporal discretization.An additional term associated with spatial discretization is added into the traditional symplectic algorithms to compensate for the numerical dispersion caused by time-discrete in numerical calculation.The series of algorithms are named as modified symplectic algorithm.In order to verify the adaptability of the modified symplectic time discretization algorithm combined with various spatial discretization methods.In this thesis,I combine modified symplectic time discretization with quasi-particle method to solve wave equation in quasi-particle system.which is call modified symplectic quasi-particled discrete method and combined modified symplectic time discretization with triangular finite element method space discretization to solve surface irregularity modeling,which is call modified symplectic scheme with finite element method.The modified symplectic algorithm is studied by theoretical analysis and numerical calculation.This thesis has done four parts research:1.Studied and summarized the advantages and disadvantages of the existing numerical methods for wave equations and the adaptation occasions.By comprehensively comparing various time-discrete methods,it is found that when the second-order symplectic algorithm is used to discretize the time parts of partial differential equations,there are advantages of less time iteration steps and high calculation accuracy.However,the traditional second-order symplectic algorithm has the problem of poor computational ability of large time steps when calculating the physical phenomena of inhomogeneous media,and finds that it has room for further improvement.The algorithm is further developed on the basis of the second-order symplectic algorithm,and the modified symplectic algorithm is derived based on the algorithm.2.The spatial discrete system of quasi-particle method is studied.The particle method is suitable for solving the calculations involving the structure of the element,electromagnetic and its coupling.Based on Hamilton mechanics,we develop a quasi-particles system to discretize seismic wave equation from the viewpoint of molecular dynamics.In this thesis,the interaction coefficients of the particles are derived.3.Aiming at the shortcomings of finite element space discrete memory consumption,the stiffness matrix storage strategy in the case of triangle discrete is studied to reduce the memory requirement in the calculation process.4.The modified symplectic algorithm constructed in this paper is combined with different spatial discrete formats in time and space,and their numerical dispersion properties and numerical stability are theoretically studied by mathematical methods.In order to simulate the infinite space in a finite model,the perfect matching layer(PML)absorption boundary condition is studied.Through theoretical analysis and specific numerical examples,it is proved that the various modified symplectic schemes proposed in this paper have the advantages of weak numerical dispersion,strong stability and good spatial discrete space-time combination performance in solving wave equations.