High Efficient Time Stepping Methods For Dynamic Equations-Algorithm Design,Analysis And Application

In the areas of civil engineering,material science,structural analysis in aerospace and electrical engineering,many key problems can be described by dynamical models.After spatial discretization,these models are reduced to systems of linear/nonlinear ordinary dif-ferential equations.This thesis is intended to develop algorithm design,analysis and appli-cation for such systems?called dynamical equations for simplicity?.Therefore,this study is of great importance in theory and application.First of all,borrowing the ideas in[1,2]for constructing adaptive time stepping method-s to solve second order evolution equations,and exploiting three local linearization methods or local quadratic approximation for discretizing the nonlinear term,we devise the adaptive C⁰P₁time stepping method and the adaptive C⁰P₂method for solving linear/nonlinear dy-namical equations.In treatment of linear dynamical equations,the a posteriori error analysis is established for the two methods,by means of the governing equation of the numerical solution as well as the theory of Lagrange/Hermite interpolation.The derivations are greatly simplified compared to the similar ones in the references[1,2].A number of numerical experiments are provided to show the reliability and efficiency of the a posteriori estimators and the two adaptive methods.Next,to improve the disadvantage that the adaptive C⁰P₂time stepping method must choose the lower bound of the mesh sizes very small to get the numerical solution with de-sired accuracy,the hp hybrid adaptive time stepping method?Algorithm 6?is proposed for solving?1.1?.The ideas of the method are very simple,that means,in the time period the exact solution varies smoothly we are tempted to use the finite element method for discretiza-tion,while in the time period the exact solution varies rapidly,we are tempted to use spectral Picard method for numerical solution.Moreover,we also use an adaptive strategy in[41]for selecting the times of the spectral Picard iteration.This method has the advantage of small amount of computational cost and high accuracy of numerical solution,and is particularly suitable for capturing the solution with rapid change.In theory,by means of a subtle anal-ysis,we obtain the hp-type convergence and error analysis for the spectral Picard method proposed by Tang et al in[3,4].The result rigorously shows the influence of the initial error,the iteration number and the polynomial order used in the collocation method on the error of the numerical solution,from which we can find that if the initial values are given exactly,this method is really an explicit and high accuracy method.This work is original and has great value of applications.Moreover,motivated by the ideas of constructing spectral deferred correction?SDC?method for first order problems,we devise a direct spectral deferred correction?d-SDC?method for solving vibrational problems without damping,by first formulating the second order equations as an integral equation of the correction error and then solving it using the usual Euler methods.Compared to the usual SDC method,the d-SDC method reduces the computational cost while can obtain much more higher accurate numerical solution.In the-ory,using a subtle analysis combined with the error estimates for the related collocation method,we achieve convergence and error analysis for the d-SDC method,indicating that the method is a high accuracy method,decreasing the error of the numerical solution rapidly with respect to the iteration number.Finally,we use the numerical methods developed in this thesis to numerically solve four problems from engineering applications,the problems including soliton simulation of sine-Gordon equations,vehicle bridge coupling vibration problems,simulation of van der Pol self oscillation systems and half plane elastic wave propagation.We present an entire procedure for studying each problem numerically,including the steps of mathematical modeling,spatial discretization,numerical solution for the dynamic equations and numerical simulation.