ee256-08-lecture03 - 1 EE256 Numerical Electromagnetics...

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1 EE256 Numerical Electromagnetics H. O. #6 Marshall 24 June 2008 Summer 2008 LECTURE 3 3.1 FINITE DIFFERENCE SOLUTIONS OF THE CONVECTION EQUATION As the simplest example of a PDE, we consider finite difference solutions of the convection equation [2.20]. As mentioned before, this equation constitutes the simplest prototype for hyperbolic equations, and is also highly relevant for our FDTD analyses of Maxwell’s equations, since it has the same form as the component curl equations that we shall solve in FDTD solutions of electromagnetic problems. Consider the convection equation written in terms of voltage on a transmission line: V ∂t + v p V ∂z = 0 [3.1] where v p 1 ft-(ns) - 1 , i.e., the speed of light in free space. As mentioned in Lecture #2, the exact solution of this convection equation is: V ( z,t ) = V + ( z - v p t ) [3.2] where V + ( · ) is an arbitrary function. A particular solution is determined by means of the initial condition at time t = 0 , i.e., V + ( z, 0) = Φ( z ) , where Φ( z ) is the initial voltage distribution on the line. The solution at any later time is then given by: V ( z,t ) = Φ( z - v p t ) [3.3] In other words, the initial voltage distribution simply is convected (or propagated) to the right at the velocity v p , preserving its magnitude and shape. We shall now consider FDTD solutions of [3.1] for a particular initial voltage distribution: V ( z, 0) = Φ( z ) = 12 z V 0 z 0 . 5ft , - 12(1 - z ) V 0 . 5ft z 1 . 0ft , 0 V Elsewhere [3.4] The exact solution is simply the triangular pulse moving to the right at the speed of v p 1 ft-(ns) - 1 , as shown in Figure 3.1. We now proceed to numerically solve for V ( z,t ) , using two different methods, namely (i) the forward- time centered-space method, and (ii) the leapfrog method.
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2 0.5 1 1.5 2 2.5 2 4 6 8 t =0 v p t =0.25 ns t =0.75 ns z (ft) V ( z,t ) (V) Figure 3.1: Exact solution of the convection equation for a triangular initial voltage distribution. The triangular distribution simply moves to the right at the speed v p 1 ft-(ns) - 1 . 3.1.1 The Forward-Time Centered-Space Method As our first approximation of the PDE given in [3.1], we use the first-order forward-difference approximation (i.e., equation [2.42]) for the time derivative and the second-order centered-difference approximation (i.e., equation [2.55]) for the space derivative. We thus have: V ∂t + v p V ∂z = 0 V n +1 i -V n i Δ t + v p V n i +1 -V n i - 1 z = 0 [3.5] We now solve [3.5] for V n +1 i to find: Forward-time centered-space V n +1 i = V n i - ± v p Δ t z ² ³ V n i +1 -V n i - 1 ´ [3.6] In comparing different finite difference methods, it is useful to graphically depict the grid points involved in the evaluation of the time and space derivatives. Such a mesh diagram is given in Figure 3.2, for both of the methods used in this section. For the solution of [3.6], we choose
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ee256-08-lecture03 - 1 EE256 Numerical Electromagnetics...

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