Superimpose these curves on your stability diagram

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Unformatted text preview: ents that sum to unity and the von Neumann method requires constant coe cient linear problems with periodic initial data. Matrix methods can be used to analyze more general linear problems. With the goal of describing these methods, consider a linear nite di erence scheme in the form of (3.1.7). For simplicity, assume that the di erence operator L is independent of the time level n and iterate this (3.1.7) to obtain Un = (L )nU0 : (If L depended on n, we would get a product of the operators at each time level.) Taking a norm kUnk = k(L )nU0k k(L )nkkU0k: The nite di erence scheme (3.1.7) is stable according to (3.1.14b) if and only if there exists a constant C such that k(L )nk C n!1 x t!0 n t T: (3.3.1) 22 Basic Theoretical Concepts Using matrix stability methods, we try to limit the growth of kL k. Before beginning, we review some material about matrix eigenvalue problems. If xi is an eigenvector of L corresponding to the eigenvalue i, then L xi = i x i i = 1 2 ::: N (3.3.2) where N is the dimension of L . De nition 3.3.1. The spectral radius (L ) of L is the modulus of its largest eigenvalue, i.e., (L ) 1miax j i(L )j: N (3.3.3) The spectral radius provides a lower bound to any matrix norm. Lemma 3.3.1. Let (L ) and kL k be the spectral radius and any vector-induced matrix norm of L , then (L ) kL k: (3.3.4) Proof. Take a norm of (3.3.2) kL xi k = j ijkxik i = 1 2 ::: N and use (3.1.10, 3.1.11) i j ij = kLxix k kk kL k i = 1 2 : : : N: The result (3.3.4) follows since this relation holds for the value of i corresponding to the maximum eigenvalue. Since the eigenvalues of (L )n are ( i)n, we can use (3.3.4) and (3.1.11) to obtain n (L ) k(L )nk kL kn: (3.3.5) The left and right sides of (3.3.4) lead, respectively, to necessary and su cient stability conditions. 3.3. Matrix Stability Analysis 23 Theorem 3.3.1. (The von Neumann necessary stability condition.) A necessary condition for the stability of the linear homogeneous nite di erence scheme (3.1.7) is that there exist a constant c, independent x and t, satisfying (L ) 1 + c t: (3.3.6) Proof. Using (3.3.5), we see that n(L ) must be bounded otherwise k(L )nk couldn't be. Thus, a necessary condition for the stability of (3.1.7) is that there exist a constant C such that n (L ) C for all n t T as x, t ! 0. Taking an nth root and using the maximal value of n gives (L ) C 1=n C t=T : When t is su ciently small, Taylor's formula can be used to nd a value of c so that C t=T is bounded by 1 + c t. Theorem 3.3.2. A su cient condition for the stability of the linear nite di erence scheme (3.1.7) is that there exist a constant c, independent of x and t, such that kL k 1 + c t: (3.3.7) Proof. The proof follows the arguments used in Theorem 3.2.1. Remark 1. The von Neumann condition (3.3.6) is also su cient for stability in many cases. For example, it is su cient for stability when L is either symmetric or similar to a symmetric matrix 2]. Example 3.3.1. In Example 3.1.4 we showed that L for the forward time-centered space scheme (2.3.3) for the heat conduction problem (2.3.1) satis es 2 1 ; 2r r 6 r 1 ; 2r r 6 r 1 ; 2r r L =6 6 6 ... 4 r 1 ; 2r 3 7 7 7: 7 7 5 If r 1=2, then kL k1 = 1 therefore, from Theorem 3.3.2, this scheme is stable in the maximum norm. 24 Basic Theoretical Concepts Example 3.3.2. Consider the nite di erence scheme Ujn+1 = c;1Ujn;1 + c1Ujn+1 with periodic initial data in j with period J . (This problem is similar to Exercise 1.5 of Strikwerda 5].) The Lax-Friedrichs scheme (3.2.7) for the kinematic wave equation (2.2.1) has this form with c; 1 = 1 + 2 c1 = 1 ; 2 where, for simplicity, we have assumed the Courant number to be constant. The di erence scheme has the form of (3.1.7) with 2 3 0 c1 c;1 6 c;1 0 c1 7 6 7 L =6 7 ... 4 5 c1 c;1 0 2 6 n U =6 6 4 3 U0n U1n 7 7 ... 7 : 5 UJn;1 If jc;1j + jc1j 1 then kL k1 1 and the scheme is stable in the maximum norm. This is the case for the Lax-Friedrichs scheme (even for nonlinear problems) when j j 1. We can analyze the stability of nite di erence schemes using the basic de nitions (3.1.14a, 3.1.14b). Nonlinear problems can be analyzed in this manner thus, in some sense, it is the most powerful analytical technique at our disposal, but it is also the most di cult one to apply. Let us try an example. Example 3...
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