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ee256-07-lecture07_2007

# ee256-07-lecture07_2007 - 1 EE256 Numerical...

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Unformatted text preview: 1 EE256 Numerical Electromagnetics H. O. #10 Inan 16 April 2007 Spring 2007 LECTURE 7 7.1 NUMERICAL DISPERSION AND DISSIPATION (DIFFUSION) The von Neumann stability condition is widely applicable and enables the assessment of the stability of any finite difference scheme in a relatively simple manner. However, such an analysis reveals little about the detailed properties of the difference scheme and especially the important properties of dispersion and dissipation (sometimes also called diffusion). We now consider the assessment of dispersive and dissipative properties of finite difference schemes. In general, a partial differential equation couples points in space and time by acting as a transfer function, producing future values of a quantity distribution, e.g., V ( x, t ) , from its initial values e.g., V ( x, 0) . The properties of a PDE or a system of PDEs can be described by means of its effects on a single wave or Fourier mode in space and time: V ( x, t ) = C e j ( ωt + kx ) [7.1] where ω is the frequency of the wave, and k is the wave number corresponding to wavelength λ = 2 π/k . By inserting [7.1] into a PDE we obtain its dispersion relation, ω = f 1 ( k ) [7.2] which relates the frequency, and thus corresponding time scale, to a particular wavelength for the phys- ical phenomenon described by the PDE. Note that, in general, the frequency ω may be real, when the PDE describes oscillatory or wave behavior, or it may be imaginary when the PDE describes the growth or de- cay of the Fourier mode, or a complex number in those cases where there is both oscillatory (wave-like) or dissipative behavior. Numerical dispersion and dissipation occurs when the ‘transfer function’ or the amplification factor of the corresponding FDE is not equal to that of the PDE, so that either phase (dispersion) or amplitude (dissi- pation) errors occur as a result of the finite difference approximation. Ideally, we assess the dispersive and dissipative properties of an FDE by obtaining the dispersion relation of the scheme, relating the frequency of a Fourier mode on the mesh to a particular wavelength λ (or wavenumber k ): ω = f 2 ( k, Δ x, Δ t ) [7.3] The applicability and accuracy of a finite difference scheme can be assessed and analyzed in detail by comparing [7.2] and [7.3]. 2 7.1.1 The Lax Method To illustrate the method, we consider once again the convection equation and the Lax method (see Lecture#6 Notes). Firstly, the dispersion relation for the PDE is simply ω = − v p k [7.4] In other words, the physical nature of the convective equation is such that ω can only be real, so that no damping (or growth) of any mode occurs and all wavenumbers have the same phase and group velocities, so that there is no dispersion. The input property distribution is simply propagated in space, without any reduction in amplitude or distortion in phase. Note that the minus sign in [7.4] is simply due the fact that we choose to use a Fourier mode with + jkx...
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