111a with b replaced by the vector z the maximum and

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Unformatted text preview: kAk1 = max i X j jaij j kAk2 = max j j (AT A)j1=2 j (3.1.12) 3.1. Consistency, Convergence, Stability 7 where j is an eigenvalue of the matrix AT A. Additional matrix norms and properties of norms appear in Strikwerda 5], Appendix A. Example 3.1.5. The maximum norm of the matrix L of (3.1.8) is kL k1 = j1 ; 2rj + 2r. If r 1=2 then (1 ; 2r) > 0 and kL k1 = 1. With these preliminary considerations established, we de ne the concepts of convergence and stability for nite di erence schemes. De nition 3.1.8. A nite di erence approximation converges to the solution of a partial di erential equation on 0 < t T in a particular vector norm if kun ; Unk ! 0 n!1 x!0 t!0 n t T: (3.1.13) Remark 5. The vector un is the restriction of the continuous solution u(x tn ) to the mesh. Remark 6. When applying De nition 3.1.8, visualize a sequence of computations performed on 0 < t T using ner-and- ner meshes. Convergence implies that the discrete and continuous solutions approach each other for t 2 (0 T ] in a particular vector norm as the mesh spacing decreases. De nition 3.1.9. Let Un and Vn satisfy homogeneous nite-di erence initial value problems with di erent initial conditions, i.e., Un+1 = L Un U0 = Vn+1 = L Vn V0 = : A nite di erence scheme is stable if there exists a positive constant C , independent of the mesh spacing and initial data, such that kUn ; Vnk C kU0 ; V0k n!1 x!0 t!0 n t T: (3.1.14a) Remark 7. The requirement that C be independent of x, t, U0 , and V0 implies that the bound expressed in (3.1.14a) is uniform. Remark 8. De nition 3.1.9, like De nition 3.1.8, can be thought of in relation to a sequence of calculations performed on ner-and- ner meshes. 8 Basic Theoretical Concepts Remark 9. This concept of stability implies that initial bounded di erences between two solutions remain bounded for nite times when the mesh spacing is su ciently small. Nothing in De nition 3.1.9 implies that C 1 hence, contrary to notions of stability that arise in physics and engineering, some growth of the di erence between solutions is permitted. Our most successful stability analyses will occur when L is linear. In this case, the matrix L is independent of Un and Vn and we have Un+1 ; Vn+1 = L (Un ; Vn) U0 ; V0 = ; : Replacing Un ; Vn by Un, we see that stability can be de ned without introducing a perturbation. This alternate de nition of stability is used throughout numerical analysis (cf., e.g., Strikwerda 5], Section 1.5) and we repeat it here for completeness. De nition 3.1.10. A nite di erence scheme (3.1.7) for a homogeneous initial value problem is stable if there exists a positive constant C , independent of the mesh spacing and initial data, such that kUnk C kU0 k n!1 x!0 t!0 n t T: (3.1.14b) Remark 10. De nitions 3.1.9 and 3.1.10 are equivalent for linear homogeneous initial value problems however, both forms are used for nonlinear problems where they are not equivalent. Remark 11. The notion of stability expressed by De nition 3.1.10 implies a bound on the growth of the solution and, as such, is similar to the concept of a \well posed" partial di erential equation. Let us use these de nitions to establish the convergence or stability of some of the nite di erence schemes that were introduced in Chapter 2. Example 3.1.6. We show that the solution of the forward time-centered space scheme (2.3.3) for the heat-conduction problem (2.3.1) converges in the maximum norm when r 1=2. Using (2.3.3a) and (3.1.2a) with v replaced by u, the nite di erence and partial di erential equation solutions satisfy Ujn+1 = rUjn;1 + (1 ; 2r)Ujn + rUjn+1 3.1. Consistency, Convergence, Stability 9 un+1 = run;1 + (1 ; 2r)un + run+1 + t jn j j j j j = 1 2 : : : J ; 1: Letting en = un ; Ujn and subtracting j j en+1 = ren;1 + (1 ; 2r)en + ren+1 + t jn j j j j j = 1 2 : : : J ; 1: Taking the absolute value and using the triangular inequality jen+1j jrjjen;1j + j1 ; 2rjjenj + jrjjen+1j + tj jn j j j j j j = 1 2 : : : J ; 1: Replacing the error terms on the right side of the above expression by their maximum values kenk1 and k nk1, we obtain jen+1j (jrj + j1 ; 2rj + jrj)kenk1 + tk nk1 j j = 1 2 : : : J ; 1: We know that r > 0. If, in addition, r 1=2, then 1 ; 2r signs can be removed from the terms involving r to obtain jen+1j kenk1 + tk nk1 j 0 and the...
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