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This leads to the von neumann condition de nition 321

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Unformatted text preview: n are stable and conversely as expressed by the following theorem. Theorem 3.2.1. A constant coe cient scalar one-level di erence scheme is stable in the Euclidean norm if and only if it satis es the von Neumann condition. Proof. Suppose the von Neumann condition is satis ed. We follow the arguments used in Example 3.2.2 and use Parseval's relation (3.2.4a) to obtain kUnk2 = J 2 J ;1 X k =0 jMk j2njA0 j2 (1 + c t)2n kU0k2: k 2 Using (3.2.5), we have kUnk2 e2cn tkU0k2 e2cT kU0k2 2 2 2 where, n t T , with T being the total time of interest. Choosing C 2 = ecT establishes stability according to (3.1.14b). 3.2. Fourier Stability Analysis 19 To prove the converse, we suppose that there is a value of k = k such that jMk j > (1 + c t) 8c: Further suppose that initial data is selected so that A0 6= 0 and A0 = 0, k 6= k . Thus, k k the initial condition is Uj0 = A0 !jk : k The solution after n time steps is Ujn = (Mk )nA0 !jk = (Mk )nUj0 : k Taking the Euclidean norm kUnk2 = jMk j2nkU0 k2 > (1 + c t)2n kU0k2 2 2 2 8c: Since this must hold for any and all values of c, kUnk2 must be unbounded and, hence, the nite di erence method is unstable. The von Neumann condition is a necessary condition for stability in much more general situations than stated by Theorem 3.2.1. Su ciency has also been veri ed under less restrictive conditions 2, 3]. Example 3.2.3. It may seem like we are restricted to using the directional derivatives when solving the kinematic wave equation (2.2.1) however, the Lax-Friedrichs scheme uses centered space di erence with a forward-averaged time di erence to obtain the approximation Ujn+1 ; (Ujn+1 + Ujn;1)=2 n Ujn+1 ; Ujn;1 + aj =0 t 2x or 1+ n 1; n Ujn+1 = 2 j Ujn;1 + 2 j Ujn+1: (3.2.7) The computational stencil for this scheme is shown in Figure 3.2.1. This spatiallycentered di erence scheme is clearly stable in the maximum norm by the Maximum Principle as long as the Courant, Friedrichs, Lewy Theorem is satis ed, i.e., as long as j jnj 1. It is also stable in L2 under similar conditions (cf. Problems 2 and 3 at the end of this section). Problems 20 Basic Theoretical Concepts 11 00 n+1 11 00 j-1 11 00 j n j+1 Figure 3.2.1: Computational stencil for the Lax-Friedrichs scheme (3.2.7). 1. Let Ujn, j = 0 1 : : : J ; 1, and An, k = 0 1 : : : J ; 1, be discrete Fourier pairs k according to (3.2.1a). Derive the discrete form of Parseval's relation (3.2.4a). 2. Consider the Lax-Friedrichs di erence scheme (3.2.7) for the constant-coe cient kinematic wave equation ut + aux = 0: 2.1. Calculate the leading terms in the local discretization error for this scheme. Is the Lax-Friedrichs scheme consistent for all choices of mesh spacings? Explain. 2.2. Use a von Neumann stability analysis to show that the Lax-Friedrichs scheme is stable in L2 whenever the Courant, Friedrichs, Lewy Theorem is satis ed 3. Consider nite-di erence schemes for the constant-coe cient kinematic wave equation of Problem 2 that have the form Ujn+1 = Ujn ; 2 (Ujn+1 ; Ujn;1) + 2 (Ujn+1 ; 2Ujn + Ujn;1) (3.2.8a) with =a t x = z: (3.2.8b) The parameter is the Courant number and and z are dissipation factors. Some common schemes having this form for speci c choices of z and appear in Table 3.2.1. The function sgn(x) = x=jxj and the Lax-Wendro scheme is described in Section 3.3. Find the region in the ( )-plane where the ampli cation factor associated with the scheme (3.2.8) does not exceed unity in magnitude. You may, 3.3. Matrix Stability Analysis 21 Method z Centered 0 Lax-Wendro Upwind sgn Lax-Friedrichs 1= 0 2 sgn 1 Table 3.2.1: Values of the dissipation factors for schemes of the form (3.2.8). if you want, con ne your analysis to positive values of . This region of parameter space is often called a \region of absolutely stability." Present a graph of the absolute stability region. The speci c methods presented in the table correspond to curves in the ( )-plane. Superimpose these curves on your stability diagram. Comment on the stability properties of the various methods. 3.3 Matrix Stability Analysis The tools that are currently available to us for performing stability analyses have severe limitations. The maximum principle (Theorem 3.1.1) requires nite di erence schemes to have positive coe ci...
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