New version of FDTD with GFM program in Schwarzschild space–time

Abstract

Finite difference time domain (FDTD) method in curved space–time is developed by filling the flat space–time with an equivalent medium. Green function Method (GFM) in curved space–time is solving transport equations. This is a new version of FDTD and GFM program in Schwarzschild space–time. The code for connection boundary and output boundary are included into the new version, so that electromagnetic scattering problem (not by black holes) in Schwarzschild space–time can be solved by this new version. New version program summaryProgram Title: FDTD with GFM in Schwarzschild space–timeProgram Files doi:http://dx.doi.org/10.17632/8fxbp578ys.2Licensing provisions: GPLv3Programming language: MATLABJournal reference of previous version: Comput. Phys. Comm. 225 (2018) 166–173Does the new version supersede the previous version?: YesReasons for the new version: Electromagnetic scattering problems cannot be solved by the old version. The code for connection boundary and output boundary are included into the new version, so that scattering problems (not by black holes) can be solved by the new version.Summary of revisions: The code for connection boundary and output boundary are included into the new version.Nature of problem: Simulating electromagnetic propagation and scattering in Schwarzschild space–time.Solution method: The electric field and the magnetic field are [1] (1)E=c(A0;1−A1;0,A0;2−A2;0,A0;3−A3;0)−−gϵ0(Ã3;2−Ã2;3,Ã1;3−Ã3;1,Ã2;1−Ã1;2)(2)Z0H=c−g(A3;2−A2;3,A1;3−A3;1,A2;1−A1;2)+1ϵ0(Ã0;1−Ã1;0,Ã0;2−Ã2;0,Ã0;3−Ã3;0)where c is the vacuum light speed, ϵ0 is vacuum permittivity, Z0 is the vacuum wave impedance, g is the determinant of metric matrix. Aα and Ãα are electric and magnetic potential respectively. The potentials can be expressed in integral form: (3)Aα(x)=μ04π∫Gαβ′(x,x′)Jβ′(x′)−g(x′)d4x′(4)Ãα(x)=ϵ04π∫Gαβ′(x,x′)J̃β′(x′)−g(x′)d4x′where μ0 is vacuum permeability, Gαβ′ is Green function, Jα and J̃α are electric and magnetic current respectively. Equation (57) in [2] has shown the expression of Aα;γ used in Eq. (1) and (2). By comparing Eq. (3) and (4) it can be seen that the expression of Ãα;γ can be obtained by replacing μ0 with ϵ0, J with J̃.The connection boundary and output boundary should be included when solving scattering problems by FDTD method [3]. The zone within the connection boundary is the total field zone, and the zone outside the connection boundary is the scatter field zone (Fig. 1). The incident wave is incorporated into FDTD simulation through connection boundary [3]. Usually the observer is very far away from the scatterer. It is impossible to simulate the whole field distribution between scatterer and observer by FDTD method. To obtain far field outside the FDTD domain, the electromagnetic current on output boundary should be integrated using Green function method which has been introduced in [2].Electric dipole is adopted as the excitation source. To reduce leakage, the distance between the positive charge and the negative charge is set to the mesh size of FDTD. At the positions of the two charges, the differential elements of electric current are (5)Jα−gd3x=(±cq,0,0,0)where q is the charge. At the midpoint of the two charges, the differential element of electric current is (6)Jα−gd3x=(0,dqdtl)where l is the vector directing from −q to q. The incident electromagnetic field on connection boundary can be calculated by Green function method which has been introduced in [2].The effective electromagnetic current ( Jα, J̃α) and their time derivatives (Jα,0, J̃α,0) on output boundary are calculated in the following way:The 3-D effective electric current J=(J1,J2,J3) and magnetic current J̃=(J̃1,J̃2,J̃3) on output boundary are: (7)J(k+1∕2)=n×H(k+1∕2)J̃(k)=−n×E(k)where n is the outer normal vector on output boundary, and k is time step. The time derivatives of 3-D current can be obtained by difference formulas (8)J,0(k)=J(k+1∕2)−J(k−1∕2)Δx0J̃,0(k−1∕2)=J̃(k)−J̃(k−1)Δx0where Δx0=cΔt is the size of time step. The time derivatives of J0 and J̃0 can be obtained by the law of charge conservation: (9)J0,0(k+1∕2)=−∇⋅J(k+1∕2)J̃0,0(k)=−∇⋅J̃(k)where the divergence of J and J̃ can be written as the difference of H and E respectively. J0 and J̃0 can be obtained by difference formulas: (10)J0(k+1)=J0(k)+Δx0J0,0(k+1∕2)J̃0(k+1∕2)=J̃0(k−1∕2)+Δx0J̃0,0(k)[1]J. L. Synge, Relativity: The General Theory, North-Holland, 1960.[2]Shouqing Jia, Dongsheng La and Xuelian Ma, Comput. Phys. Comm. 225 (2018) 166–173.[3]Atef Z. Elsherbeni and Veysel. Demir, The Finite-Difference Time-Domain Method For Electromagnetics with MATLAB Simulations, 2nd Ed, SciTech Publishing, 2015.