Stability Analysis Of Perfectly Matched Layer Methods

Perfectly Matched Layer (PML) method is one of the most prevalent and effectivemethod in computational electromagnetics nowadays,and the thesis is to studythe stabilities of PML methods.Particularly,for the differential system,we collect the stability results proved byenergy methods;for the difference system,we utilize Fourier methods to analysis theYee scheme of Cauchy problem of B(?)renger PML,and prove that the system is onlyspectrum stable under certain mesh size conditions.Also,the Yee scheme of UPMLsystem satisfies the spectrum stability with the same mesh size conditions as thoseof B(?)renger PML,moreover,the scheme of UPML system with same damping bothsides is stable.Spectrum stability is a kind of weak stability.In order to establishthe foundations of the stability of PML,we use energy methods to study the initialboundary value problem of Maxwell equations truncated by square UPML,andprove that the FDTD system in the truncated domain is stable.Similar results areobtained for circular UPML and polar coordinate cases.Numerical experiments arecarried out to verify the stability results.The innovations of the thesis are:1.By defining a concept of spectral stability,and using Miller-Schur criterion to analyze the amplification matrices with softwareMathematica,we get the mesh size conditions of the Yee schemes of the Cauchyproblem of the PMLs;2.It happens to find that the spectral stability mesh sizeconditions of PML and those of UPML are the same,and the spectral stabilityof the PML can not be improved,however the UPML scheme with same dampingboth sizes is stable;3.In order to prove the stability of the initial boundary valuesystem of the UPML in the truncated domains,we deduce some equivalent systemsby introducing proper variables changes and define some Taylor parameters.Byvirtue of the equivalent systems and Gronwall inequality,we manage to finish theproof.The significance of the thesis is to collect and prove the stabilities of the PMLs,and to construct the numerical theory foundation of the PMLs to Maxwell equations.These results enhance the cognitions to the stabilities of PMLs and can be relied onin computations.