Nhng tin b trong Quang hc, Quang ph và ng dng VI ISSN 1859 - 4271

Những tiến bộ <strong>trong</strong> <strong>Quang</strong> học, <strong>Quang</strong> <strong>ph</strong>ổ và Ứng dụng VI ISSN 1859 - 4271The classical five-point stencil for the 2-D Helmholtz equation at point ucis given by1uc =2 2( ur + uu + ul + ud), k = 2 π λ .4 − k ∆(1)The local field u( ρ, φ ) near u is given in a polar coordinate system as the following FouriercBessel series:u( ρ, φ) ≈ a0J 0( kρ) + a1 J1( kρ)cos φ + b1 J1( kρ)sinφ+ a J ( kρ) cos 2 φ + b J ( kρ)sin 2 φ.2 2 2 2We wish to express u in terms of its four neighboring points. But we have five unknownscand only four known points. Note that the second J ( kρ)sin 2φcoefficient 2b2does notcontribute to the field at those neighboring points. Thus we have the following 4 by 4 matrixequation rela<s<strong>trong</strong>>tin</s<strong>trong</strong>>g the coefficients with the four given points.⎡J ( k∆) J ( k∆) 0 J ( k∆)⎤ ⎡a⎤ ⎡u⎤0 1 2 0 r⎢J0 ( k ) 0 J1( k ) - J2 ( k )⎥ ⎢a⎥ ⎢1 u⎥⎢∆ ∆ ∆⎥ ⎢ ⎥ = ⎢ u ⎥(3)⎢J0 ( k∆) - J1( k∆) 0 J2( k∆)⎥ ⎢b1⎥ ⎢ul⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ J0 ( k∆) 0 - J1( k∆) - J2 ( k∆)⎦ ⎣ a2⎦ ⎣ ud⎦Here ∆ is the horizontal and vertical spacing between points. We can solve for the a0coefficient to obtain the new FD-like five-point stencil :ucu + u + u + u= a0=4 J ( k∆)r u l d0.Next, we include four more points from the four corners to Equation (1).(2)(4)They areune, unw, usw , use,the north-eastern, north-western, south-western and south-eastern pointurespectively. We show that (detail to be published) from Fourier-Bessel series centered on thecpoint, one may obtain the general nine-point stencil for the 2-D Helmholtz equation:ucd ⋅ ( ur + uu + ul + ud ) + c ⋅ ( une + unw + use + usw)=4( a ⋅ c + b⋅d )0 0 4 4(LFE-9)a = J ( k∆ ), b = J ( 2 k∆ ), c = J ( k∆ ), d = J ( 2 k∆).III. RESULTS AND DISCUSSIONThis FD-like nine-point stencil (LFE-9) possess superior numerical properties that it has verylow temporal and spatial dispersion as depicted in Figure 2 where we compute, numerically,plane wave propagation constants κ (shown in filled circles) as function of propagationdirection. The normalized spatial sampling density N λ is defined as number of sampling pointsper wavelength, ( @ λ ∆).N λ(5)89