Energy-preserving Numerical Methods For Several Types Of Fractional Wave Equations And Maxwell's Equations

Most of the analytical solutions of fractional partial differential equations are u-nattainable,so that attempts to obtain accurate,stable and efficient numerical solutions are very necessary and important.Over the past few decades,many excellent numer-ical methods have been developed for solving fractional partial differential equation-s,such as finite difference methods,finite element methods,finite volume methods,spectral methods,and so forth.Numerical methods that could preserve some intrinsic properties of the original system,have remarkable superiority in terms of the stabil-ity,long time simulations,damping non-physical oscillation.However,it is usually difficult to construct the associated numerical methods.This thesis is mainly devot-ed to developing energy conservative numerical methods for solving several types of fractional wave equations and Maxwell's equations,i.e.,fractional Klein-Gordon-Zakharov system,nonlinear space fractional Boussinesq equation,strongly coupled nonlinear damped space fractional wave equations.The thesis includes five parts.The concrete research jobs are given as follows.In Chapter 1,we briefly introduce the definitions and properties of fractional derivatives,research background and contents.In Chapter 2,we propose an energy conservative linear difference scheme for the fractional Klein-Gordon-Zakharov system.By virtue of the discrete energy method and a "cut-off" function technique,it is shown that the proposed scheme possesses the convergence rates of O(?t2+h2)in the sense of L?-and L2-norms,respectively,is unconditionally convergent and stable.Numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results.Chapter 3 focuses on the development and analysis of an energy-preserving split-ting difference scheme for the fractional-in-space Boussinesq equation.Introducing the potential function v via(?)the original problem is transformed into the equivalent fractional parabolic equations.An energy conser-vative Crank-Nicolson scheme is proposed for solving the resulting equations.By utilizing the discrete energy method,it is shown that the proposed scheme attains the convergence rates of O(?t2+h2)in the discrete L?-norm without any restrictions on the grid ratio.A new linearized iterative algorithm is proposed to the implementation of the proposed scheme.In Chapter 4,two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed,which involve a two-level nonlinear difference scheme,and a three-level linear difference scheme based on invariant energy quadratization formulation.Then the discrete energy dissipation properties,solvability,unconditional convergence and stability of the proposed schemes are exhibited rigidly.By the discrete energy analysis method,it is rigidly shown that the proposed schemes enjoy the unconditional convergence rates of O(?t2+h2)in the discrete L?-norm for numerical solutions uk and vk,respectively.Some numerical results are provided to illustrate the physical behaviors and unconditional energy stability of the suggested schemes,and testify the correctness of theoretical results.Chapter 5 is aimed at developing and analyzing two types of new energy-preserving local mesh-refined splitting finite difference time-domain(EP-LMR-S-FDTD)schemes for two-dimensional Maxwell's equations.For the local mesh refine-ments,it is a challenging task to define the suitable local interface schemes which can preserve energy conservation and guarantee the high accuracy.The important feature of this chapter is that we propose the efficient local interface schemes on the interfaces of coarse grids and fine grids that ensure the energy conservation property,keep spatial high accuracy and avoid oscillations.And meanwhile,we propose a fast implemen-tation of the EP-LMR-S-FDTD schemes,which overcomes the difficulty in solving the "trifuecate structure".We prove the EP-LMR-S-FDTD schemes to be energy con-servative and unconditionally stable.Furthermore,we obtain the convergence of the proposed schemes.Numerical experiments are given to show the high-performance of the EP-LMR-SFDTD schemes which further confirm the theoretical results.