Splitting Energy-preserving Schemes For Three-dimensional Maxwell Equations

This thesis has proposed four energy-conserved splitting schemes for the three dimensional Maxwell equations.In order to avoid solving a large scale of algebraic equations,I use local one-dimensional(LOD)method to split the original equations into six one-dimensional subsystems,and divide the Maxwell equations into three parts according to the spatial derivatives(?)x,(?)y,(?)z.And then,I use the Lie-Trotter and Strang splitting methods.Finally,for each LOD equation,I use Crank-Nicolson(C-N)to finish time discretization,on the other hand,I use high order compact(HOC)to finish spatial discretization.Afterwards,I have proved the stability of numerical schemes as well as conservation of energy and convergence.Numerical results show that these schemes are not only stable and efficient but also energy-preserving.The structure of this thesis is as followed:Chapter 1 has introduced the physical background of Maxwell equations,related research status at home and abroad and some notations and theorems which are frequently used in this thesis.After that,I decompose the Maxwell equations into six local one-dimensional issues by LOD method.Next,I separate the Maxwell cquations into three parts according to the spatial derivatives(?)x,(?)y,(?)z.Finally,Lie-Trotter and Strang splitting methods for maintaining conservation of energy are introduced.Chapter 2 mainly studies high order compact method and its construction idea which is utilized to construct two high-precision numerical methods.Chapter 3 aims to reduce the size of algebraic equations,turns the three dimensional problems into several local one dimensional problems by two splitting methods mentioned in Chapter 1.And then,in the time direction,it is discretized by the C-N scheme,meanwhile,in the spatial direction,it is discretized by the high-order compact scheme mentioned in Chapter 2.Above all,I have obtained four numerical schemes of Maxwell equations with improved accuracy and conservation of energy.Chapter 4 is going to analyze the stability of numerical schemes,conservation of energy law and convergence,the typical properties of the new schemes.After which,I have proved that the numerical schemes are unconditionally stable and also maintain the conservation of energy,surely,they can respectively converge to time order p and space order q,where p=1 or 2;q=4 or 8Chapter 5 has simulated the electromagnetic wave described by the three-dimensional Maxwell equations by utilizing schemes constructed in Chapter 3.Finally,such schemes perfectly verified the theoretical results obtained in Chapter 4 including convergence,stability and conservation of energy.