Re: FDTD method to solve Maxwell equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg128583] Re: FDTD method to solve Maxwell equations*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Wed, 7 Nov 2012 00:57:35 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <k6t7p7$1m3$1@smc.vnet.net> <k6vkmi$6io$1@smc.vnet.net> <k773st$4gp$1@smc.vnet.net>

Am 05.11.2012 02:14, schrieb fc266 at st-andrews.ac.uk: > On Friday, November 2, 2012 5:13:46 AM UTC, Roland Franzius wrote: >> >> To implement a NDSolve method in 4 space time dimensions for the Maxwell >> >> second rank tensor field >> >> (t,x)-> F_ik(t,x) >> >> with six components obeying the constraints of exterior differential forms >> >> >> >> Dt[Wedge[F_ik Dt[xi], Dt[xk]] = 0 >> >> >> >> is probably a very ambitious project and not so much a suitable working >> >> field to learn the application of Mathematica to real space-time physics. >> >> >> >> In the present situation the given Mathematica NDSolve-methods can not >> >> handle such monster problems, monsters with respect to memory and time. > > I would just be interested in solving for the 1D and 2D case so I can simulate some basic examples, how would I be able to do that? > is that simpler? > Warning: Exterior calculus stuff. As algebraists and geometers know, in 2-d space-time you have one antisymmetric tensor field, the "volume" 2-form F = f[t,x] dct /\ dx with one single pseudoscalar density component f. The first Maxwell equation is dF=0 which is satisfied by any suitable f since any 3-form is null identically. Given the Lorentz metric g=DiagonalMatrix[{1,-1}], the Hodge dual stress tensor field S=*F is a scalar density given by the scalar product with the 2-volume density S = < cdt/\ dx , f(t,x) cdt /\dx > = <cdt,cdt> <dx,dx> f(t,x) = g^tt g^xx f(t,x) Consequently, with metrics g^tt=1, g^xx=-1 the source equation gives the dual of the source "current-charge density" vector field j dS = df/dct dct + df/dx dx = *j with df/dx = j_t/c = rho(t,x) df/dct = j_x (t,x)/c which because of ddS=0 is a conserved quantity d rho/dt + dj/dx = 0 The distributional Coulomb field solution for a fixed point charge q in the orgin is provided by the unit step function f[t_,x_]:= q/2 (UnitStep[x]-UnitStep[-x]) It can be transformed to a moving charge by a Lorentz boost x = Cosh[u] x' + Sinh[u] t' t = Sinh[u] x' + Cosh[u] t' F' = q/2 (UnitStep[#]-UnitStep[-#]&)[Cosh[u] x' + Sinh[u] t'] (Sinh[u] dx' + Cosh[u] dt')/\(Cosh[u] dx' + Sinh[u] dt') = q/2 (UnitStep[#]-UnitStep[-#]&)[(x'+v/c*t')/Sqrt[1-(v/c)^2]]* dt' /\ dx' The vacuum wave solutions are superpositions of any genereralized functions, which are constant along the right or left lightlike directions F= (f_r(x-ct) + f_l(x+ct)) dt/\dx This is the trivial content of 2-d electrodynamics. In three dimansions the Maxwell field is F = E_x dt/\dx + E_y dt/\dy + B dx/\dy and the potential group of Maxwell yields dF=0 : dE_x/dy -dE_y/dx + dB/dct = 0 As usual, any 1-Form A is a potential for a F d (A_t dct + A_x dx + A_y dy) = F with ddA = dF= 0 The stress tensor is (modulo signs?) S = < dct /\ dx /\ dy , F > = E_x dy - E_y dx + B dt so the source part of Maxwells equations are as usual dS = *j= (dE_x/dct - dB/dy) dct/\dy - (dE_y/dct + dB/dx) dct/\dx + dE_x/dx + dE_y/dy dx/\dy giving div E=rho dB/dy - dE_x/dct = j_x dB/dx - dE_y/dct = j_y Read B as B_z orthogonal to a plane to see its meaning. With this basic knowledge on can introduce Maxwell electrodynamics eg on a sphere, a torus or inside a capacitor. Since everything boils down to eigenfunction expansions and Fourier synthesis and solving with retarded Greens functions, its a nice field to play around on both sides, numerically and algebraically, using space-time NDSolve methods and series of special orthogonal function systems. After some experiments one has a chance to become an expert in one of the most fundamental physical sciences, the trival science of the vacuum. -- Roland Franzius

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**Re: FDTD method to solve Maxwell equations**

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