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# This method was introduced by john von neumann and is

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Unformatted text preview: c in j with period J and express its solution as the discrete complex Fourier series U= n j J ;1 X k =0 An!jk k j = 0 1 ::: J ; 1 (3.2.1a) where !j e2 ij=J : (3.2.1b) The complex form of the Fourier series is much more convenient for our present purposes than the more common representation in terms of sines and cosines. In this form, the solution Ujn , j = 0 1 : : : J ;1, is an approximation of the solution of a partial di erential equation that is periodic in x with period 2 . The mesh spacing that is inferred by (3.2.1a) is x = 2 =J . 3.2. Fourier Stability Analysis 13 The discrete Fourier series has many properties in common with the in nite Fourier series or the continuous Fourier transform (cf. Gottlieb and Orszag 1] or Strikwerda 5], Section 2.1). For example, the discrete Fourier series satis es the orthogonality relation J ;1 X ik l !jk !jl = J oftherwise mod J 0 j =0 (3.2.2) where k l mod J means that k and l di er by an integral multiple of J and a superimposed bar denotes a complex conjugate, e.g., !j = e;2 ij=J . The relationship (3.2.2) is easily established using properties of the complex roots of unity. Given the solution Ujn, j = 0 1 : : : J ; 1, we can nd the Fourier coe cients An, k k = 0 1 : : : J ;1, by inverting the discrete Fourier series using the orthogonality relation (3.2.2). Thus, multiplying (3.2.1a) by !jl and summing over j yields J ;1 X j =0 U!= nl jj J ;1 J ;1 X lX j =0 !j k =0 An!jk : k Interchanging the order of the summations on the right side and using (3.2.2) gives J ;1 X j =0 U!= n j l j J ;1 J ;1 X nX k =0 Ak Thus, 1 An = J l j =0 J ;1 X j =0 !jk !jl = JAn: l Ujn!jl : (3.2.3) The inverse (3.2.3) of the discrete Fourier series (3.2.1a) is called the discrete Fourier transform. Another property of discrete Fourier series and transforms that follows directly from the orthogonality condition (3.2.2) is the discrete form of Parseval's relation (cf. Problem 1 at the end of this section) kUnk2 = J kAnk2 2 2 where, for complex quantities, kUnk2 2 J ;1 X j =0 Ujn Ujn kAnk2 2 (3.2.4a) J ;1 X k =0 AnAn: kk (3.2.4b) 14 Basic Theoretical Concepts Given the link between physical and Fourier coe cients expressed by Parseval's relation (3.2.4a), it is natural to use von Neumann's stability analysis in the Euclidean (L2) norm. Let us illustrate the technique with some examples. Example 3.2.1. We use von Neumann's approach to show that the forward timen backward space nite di erence scheme (2.2.4) is stable in L2 when j is a positive constant, say, 2 (0 1] and periodic initial data is applied. In this case, (2.2.4) becomes Ujn+1 = (1 ; )Ujn + Ujn;1: Substitute (3.2.1a) into the above expression to obtain J ;1 X k =0 A J ;1 +1 ! k = X (1 ; )An ! k + j kj k =0 n k According to (3.2.1b), !j;1 = !j e;2 J ;1 X k =0 i=J An!jk;1]: k thus, An+1 ; (1 ; )An ; Ane;2 k k k ik=J !jk = 0: Using the orthogonality relation (3.2.2) and the discrete Fourier transform (3.2.3), we infer that the (bracketed) coe cient of !jk must vanish for each k, i.e., An+1 ; (1 ; ) + e;2 ik=J ]An = 0: k k Solving for An+1 k An+1 = Mk An k k Mk = (1 ; ) + e;2 ik=J : The ampli cation factor Mk gives the growth or decay of the kth Fourier mode in one time step. The above recurrence can be iterated to obtain An = (Mk )nA0 k k where the Fourier coe cient A0 , k = 0 1 : : : J ; 1, can be determined from the prek scribed initial data upon use of (3.2.3). Explicit determination of A0 is rarely necessary k since we seek to establish stability independently of the initial data. 3.2. Fourier Stability Analysis 15 We can use Eulers identity ei = cos + i sin to determine the magnitude of the ampli cation factor as jMk j2 = Mk Mk = (1 ; + cos 2J k )2 + ( sin 2J k )2 or The half-angle formula jMk j2 = 1 ; 2 (1 ; )(1 ; cos 2J k ): sin2 =2 = (1 ; cos )=2 can be used to simplify the above relation to jMk j2 = 1 ; 4 (1 ; ) sin2 Jk : We now see than the initial Fourier mode A0 will grow or decay depending on whether k jMk j is greater than or less than unity. In this example, no Fourier mode grows since 0 4 (1 ; ) 1 when 0 < 1. Using (3.2.1a), we see that the nite di erence solution satis es U= n j J ;1 X k =0 (Mk )nA0 !jk : k Stability in the Euclidean norm follows by applying Parseval's relatio...
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