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# In this example no fourier mode grows since 0 4 1 1

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Unformatted text preview: n (3.2.4a) to the above equation to obtain kU n Since jMk j nd J ;1 2 = J X jMk j2n jA0 j2 : k2 k k =0 1 for all k when 0 < kUnk2 J 2 1, we can use Parseval's relation once more to J ;1 X jA0 j2 = kU0k2: k 2 k =0 2 stability of (2.2.4) when the Courant number is a constant This establishes the L 2 (0 1]. Example 3.2.2. Let us, at long last, use von Neumann's method to examine the L2 stability of the forward time-centered space scheme (2.2.5) for the kinematic wave 16 Basic Theoretical Concepts n equation (2.2.1). Necessarily, we assume that j is a constant and the initial data is periodic in j with period J so that (2.2.5) becomes Ujn+1 = Ujn ; 2 (Ujn+1 ; Ujn;1): Expanding Ujn in a discrete Fourier series (3.2.1a), we nd J ;1 X k =0 An+1 ; An + 2 (e2 ik=J ; e;2 ik=J )An]!jk = 0: k k k Once again, the orthogonality relation (3.2.2) may be used to infer that An = (Mk )nA0 k k where Mk = 1 ; 2 (e2 ik=J ; e;2 ik=J ) = 1 ; i sin 2 k : The magnitude of the ampli cation factor is J jMk j2 = 1 + 2 sin2 2J k : There is no possibility of restricting so that jMk j 1 for all k thus, most Fourier modes will grow. As bad as this appears, (2.2.5) is still stable according to either (3.1.14a) or (3.1.14b). To demonstrate this, observe that jMk j2 1 + 2: The following inequality is useful in many situations, so we'll record it as a lemma. Lemma 3.2.1. For all real z 1+z ez : Proof. Using Taylor's formula for the exponential function ez = 1 + z + 1 e z2 1 + z: 2 (3.2.5) 3.2. Fourier Stability Analysis 17 In the present case, (3.2.5) can be used to bound the ampli cation factor as jMk j2 e : 2 Taking the nth power and using the de nition of the Courant number (2.2.3) jMk j2n ena 2 t2 = x2 = e(n t)(a2 t= x2 ): As usual, introduce T so that n t T . Also, let us agree to select meshes having temporal and spatial steps satisfying a2 t= x2 = , where is a constant. Then jMk j2n eT = C 2 : Using Parseval's relation (3.2.4a) and following the steps used in Example 3.2.1, we nd kU n J ;1 2 = J X jMk j2n jA0 j2 k2 k k =0 C 2kU0 k2: 2 We have found a constant C satisfying kUnk2 C kU0k2 hence, (2.2.5) is stable in L2. The constant C is greater than unity, so that most Fourier modes will grow. In fact, if T and are not small enough, initial disturbances will grow to such an extent that they dominate the computation. Additionally, we are forced to restrict the time step t x2 =a2, while the directional methods (2.2.2b, 4) only require t x=a. This leads us to conclude that, while technically stable, the forward time-centered space scheme (2.2.5) is not a practical method to apply to the kinematic wave equation (2.2.1). When applying von Neumann's method we will always nd a solution of the form Ujn = J ;1 X k =0 (Mk )nA0 !jk : k In Example 3.2.1, we found that jMk j 1 so that initial perturbations did not grow and the forward time-backward space scheme was stable in L2. In Example 3.2.2, jMk j > 1 and perturbations grew, but were bounded for nite times T . While the (forward timecentered space) scheme was stable, we saw that it was impractical and, thus, may ask 18 Basic Theoretical Concepts whether or not we should keep jMk j 1 as a practical matter. The answer to this question depends on the behavior of the problem under investigation. If the solution of the partial di erential equation is unstable and, hence, growing in time, then some growth of initial perturbations can and should be tolerated. However, if the solution of the partial di erential equation does not grow, then either jMk j must be bounded by unity or we must be willing to compute for su ciently small times so that the bound C on the growth of kUnk remains small. There are only a few instances where the latter option will be desirable thus, we should typically maintain jMk j 1 whenever solutions of the partial di erential equation under study do not grow. In those cases where jMk j may exceed unity, we may ask the amount by which it may do so. This leads to the von Neumann condition. De nition 3.2.1. A nite di erence scheme satis es the von Neumann condition if there exists a positive constant c that is independent of t, x, and k which satis es jMk j 1 + c t 8t t x x: (3.2.6) Finite di erence schemes that satisfy the von Neumann conditio...
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