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ee256-08-lecture06

ee256-08-lecture06 - 1 EE256 Numerical Electromagnetics...

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1 EE256 Numerical Electromagnetics H. O. #6 Marshall 24 June 2008 Summer 2008 LECTURE 6 6.1 NUMERICAL STABILITY OF FINITE DIFFERENCE METHODS The most commonly used procedure for assessing the stability of a finite difference scheme is the so-called 1 von Neumann method, initially developed (like many other finite difference schemes) for fluid dynamics related applications. The von Neumann method is based on obtaining the exact solution of the Finite Difference Equation (FDE) for a general spatial Fourier component of the complex Fourier series representation of the initial spatial distribution of the physical property (e.g., voltage) involved. If the exact solution of the FDE for the general Fourier component is bounded (either under all conditions or subject to certain conditions on Δ x , Δ t ) then the FDE is said to be stable . If the solution for the general Fourier component is unbounded, then the FDE is said to be unstable. Determining the exact solution for a Fourier component is equivalent to finding the error amplification factor q (or for a system of FDEs the error amplification matrix [ q ] ) in a manner similar to that which was done for ODEs in Lecture #2. Consider an arbitrary voltage distribution V ( x, t ) to be expressed in terms of a complex Fourier series in space, i.e., V ( x, t ) = X m = -∞ V m ( x, t ) = X m = -∞ C m ( t ) e jk m x [6.1] where C m ( t ) is dependent only on time, j = - 1 and k = 2 π/λ is the wavenumber corresponding to a wavelength λ . A general spatial Fourier component is thus given by V m ( x, t ) = C m ( t ) e jk m x [6.2] For stability analysis, we work with the discretized version of [6.2] given by V n i = C ( t n ) e jk i Δ x [6.3] 1 This stability analysis based on spatial Fourier modes was first proposed and used by J. von Neumann during World War II at Los Alamos National Laboratory [J. von Neumann, Proposal and analysis of a numerical method for the treatment of hydrodynam- ical shock problems, Nat. Def. Res. Com., Report AM-551, 1944; J. von Neumann and R. D. Richtmeyer, A method for numerical calculation of hydrodynamic shocks, J. Appl. Phys. , 21, p.232, 1950].
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2 where we have dropped the subscript m since we shall only work with a single component from here on. Note that C ( t n ) is a constant. Note that the values of V n i ± 1 are simply related to V n i as V n i ± 1 = C ( t n ) e jk ( i ± 1)Δ x = C ( t n ) e jk i Δ x | {z } V n i e ± jk Δ x = V n i e ± jk Δ x [6.4] The basic procedure for conducting a von Neumann stability analysis of a given FDE involves the following steps: 1 Substitute the discretized single Fourier component as given by [6.3] and its shifted versions as given by [6.4] into the FDE. 2 Express e jk Δ x in terms of sin( k Δ x ) and cos( k Δ x ) and reduce the FDE to the form V n +1 i = q V n i , so that the amplification factor q (or the amplification matrix [ q ] ) is determined as a function of sin( k Δ x ) and cos( k Δ x ) and Δ t .
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