Splitting Schemes For Two-dimensional Maxwell-Debye Equations

This thesis proposes two splitting schemes for the two-dimensional Maxwell-Debye equations.In order to avoid solving a large scale of algebraic equations,time-splitting method is used to split the original equations into some sub-equations,and divide the Maxwell-Debye equations into three parts according to the spatial deriva-tives(?)x,(?)y and an ordinary differential equation.And then,I use the Lie-Trotter and Strang splitting methods.Finally,for each sub-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-dissipation.The structure of this thesis is as followed:Chapter 1 has introduced the physical background of Maxwell-Debye equations,related research status at home and abroad and some notations and theorems which are frequently used in this thesis.Chapter 2 mainly studies high order compact method and its construction idea which is utilized to construct a compact method with fourth-order spatial accuracy.And then,I introduce the time-splitting method for two dimensional Maxwell-Debye equations.Finally,Lie-trotter and Strang splitting methods for maintaining dissi-pation of energy are introduced.Chapter 3 according to two splitting methods mentioned in Chapter 1,in the time direction,it is discretized by the C-N scheme,meanwhile,in the spatial di-rection,it is discretized by the high-order compact scheme mentioned in Chapter 2.Above all,I have obtained two numerical schemes for maintaining dissipation of energy of two dimensional Maxwell-Debye equations.Chapter 4 is going to analysis the stability of numerical schemes,dissipation of energy law and convergence.After which,I have proved that the numerical schemes are unconditionally stable and also maintain the dissipation of energy,surely,they can respectively converge to time order p and space order q,where p=1 or 2;q=4.Chapter 5 has simulated the electromagnetic wave described by the two di-mensional Maxwell-Debye 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.