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If in addition r 12 then 1 2r signs can be removed

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Unformatted text preview: absolute value j = 1 2 : : : J ; 1: Since this result is valid for j 2 1 J ; 1], it holds for the particular value of j where the left side attains its maximum value. Thus, we can replace the left side absolute value by its maximum norm to obtain ken+1k1 kenk1 + tk nk1: Iterating the inequality over n ken+1k1 ken;1k1 + t(k nk1 + k Thus, where ;1 k n 1) ::: : kenk1 ke0 k1 + n t = 0<k<n;1 k k k1: max Neglecting round o errors, solutions of the partial di erential and di erence equations both satisfy the same initial conditions, so ke0 k1 = 0. Additionally, if T is the total time of interest, n t T for all combinations of n and t. Thus, kenk1 T : 10 Basic Theoretical Concepts Using (3.1.2c) we bound as tK + 2 where Thus, K = 0 t max x 1 juttj T0 x2 M 12 M = 0 t max x 1 juxxxxj: T0 2 kenk1 T ( 2t K + 12x M ): Letting x and t approach zero, we see that kenk1 ! 0 hence, the forward time- centered space scheme (2.3.3) converges to the solution of the heat-conduction problem (2.3.1) in the maximum norm when r 1=2. Example 3.1.7. We show that the forward time-backward space scheme (2.2.4) Ujn+1 = (1 ; jn)Ujn + jnUjn;1: for the kinematic wave equation (2.2.1) is stable in the maximum norm when the Courant n number j a(Ujn ) t= x 2 (0 1]. (Recall that this condition must be imposed to satisfy the Courant, Friedrichs, Lewy Theorem 2.2.1.) We'll use the stability de nition (3.1.10) and begin by taking the absolute value of the di erence scheme and using the triangular inequality to obtain jUjn+1 j j1 ; jnjjUjnj + j jnjjUjn;1j: n Since 0 < j 1 by the Courant, Friedrichs, Lewy Theorem, we can remove the absolute n value signs from the terms involving j to get jUjn+1j (1 ; jn)jUjnj + jnjUjn;1j: Replacing the solution values on the right side by their maximum values jUjn j kUnk1 8j: As in Example 3.1.6, the inequality must hold for the value of j that maximizes its left side thus, kUn+1 k1 kUnk1: 3.1. Consistency, Convergence, Stability 11 This inequality holds for all n and may be iterated to yield kUnk1 kU0k1 which establishes stability in the sense of (3.1.14b). In the two previous examples, we analyzed the convergence and stability of nite di erence schemes using very similar arguments. These arguments can be generalized further to obtain a \maximum principle." Theorem 3.1.1. A su cient condition for stability of the one-level nite di erence scheme Ujn+1 = X jsj S csUjn+s in the maximum norm is that all coe cients cs , jsj S , be positive and add to unity. Proof. The arguments follow those used in Examples 3.1.6 and 3.1.7. Remark 12. A one-level di erence scheme is one that only involves solutions at time levels n and n + 1. A multilevel di erence scheme might involve time levels prior to n as well. Typically, we not only want to know that a given scheme converges but the rate at which the numerical and exact solutions approach each other. This prompts the notion of order of accuracy. De nition 3.1.11. A consistent nite-di erence approximation of a partial di erential equation is of order p in time and q in space if n j = O( tp) + O( xq ): (3.1.15) Problems 1. Using (3.1.2b), we easily see that the forward time-centered space scheme (3.2.3a) for the heat conduction equation is rst order in time and second order in space. Show that this scheme has an O( x2) local discretization error when r = t= x2 = 1=6. 12 Basic Theoretical Concepts 2. Show that the Euclidean norm of a J J matrix A given by (3.1.12) follows from from the de nition of a matrix norm (3.1.10) upon use of the Euclidean vector norm (3.1.9b). 3. In Example 3.1.7, we used the maximum principle (Theorem 3.1.1) to establish stability of the forward time-backward space scheme (2.2.4). It can likewise be used to establish stability in the maximum norm of the forward time-forward space n scheme (2.2.2b) when ;1 j 0. Can it be applied to the forward time-centered space scheme (2.2.5)? 3.2 Stability Analysis Using a Discrete Fourier Series A discrete Fourier series can be used to analyze the stability of constant coe cient nitedi erence problems with periodic initial data in much the same way that an in nite Fourier series can be used to solve constant coe cient partial di erential equations. This method was introduced by John von Neumann and is called von Neumann stability analysis. Thus, suppose that the solution of a nite di erence problem is periodi...
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