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As necessary we assume that the initial conditions

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Unformatted text preview: ourier series (3.2.1), i.e., J ;1 X nk n= Uj Ak !j !j = e2 ij=J : (4.1.16) k=0 Substituting (4.1.16) into (4.1.7b) J ;1 X n+1 fAk ;r e;2 k=0 ik=J + 1 + 2r ; r e2 ik=J ]; 4.1. Implicit Di erence Methods An r(1 ; )e;2 k ik=J 9 + 1 ; 2r(1 ; ) + r(1 ; )e2 ik=J ]g! k j = 0: Using the orthogonality property (3.2.2) and Euler's identity, we nd An+1 = Mk An k k where the ampli cation factor Mk satis es 2 ; 2k =J ) Mk = 1 ; 1 + r(1 (1 coscos 2k =J ) : 2r ; Using the half angle formula 2 Mk = 1 ; 4r sin k2 =J : 1 + 4r sin k =J (4.1.17) As usual, we can solve the rst-order di erence relation for An to get k An = (Mk )nA0 : k k (4.1.18) Since solutions of periodic initial value problems for (4.1.1) are decaying, let us demand that no Fourier mode grow thus, jMk j 1 for all k. Since Mk is real and r and are non-negative, this implies 2 4 sin k =J ;1 1 ; 1 + r r sin2 k =J 1: 4 The right-hand inequality is always satis ed hence, it remains to show that 4r sin2 k =J 1 + 4r sin2 k =J or 2 2r(1 ; 2 ) sin2 kJ 1: Since this must be satis ed for all k, it su ces to restrict 2r(1 ; 2 ) 1: (4.1.19) (This restriction is also necessary since sin k =J will be unity for some choices of k and J .) 10 Parabolic PDEs If 1=2 1, the term 1 ; 2 will be non-positive, and (4.1.19) will be satis ed for all values of r > 0. Thus, (4.1.7b) is stable for all choices of t and x. We call such a scheme unconditionally stable. If 0 < 1=2, then jMk j 1 when r 1 2(1 ; 2 ) : (4.1.20) As in Section 3.2, we may use (4.1.18) in (4.1.16) to nd the solution of (4.1.7b) as Ujn = J ;1 X k=0 (Mk )nA0 !jk : k (4.1.21) When jMk j 1, we use Parseval's relation (3.2.4) to show that kUnk2 kU0k2. In Chapter 3, we noted that it was essential to maintain jMk j 1 whenever the solution of the partial di erential equation does not increase in time. For any particular problem, initial conditions may be prescribed such that A0 = 0 whenever jMk j > 1. It k would appear that jMk j could exceed unity in this case however, round o errors will excite this mode and the growth of jMk j will eventually dominate the solution. These arguments motivate the need for an alternate stability de nition. De nition 4.1.1. A nite di erence scheme Un+1 = L Un (4.1.22a) is absolutely stable for a given mesh (of size x and t) if kUnk kU0k n > 0: (4.1.22b) Remark 1. This de nition parallels (3.1.14b). A similar one can be stated in the terminology of (3.1.14a). Remark 2. The basic stability de nitions (3.1.14a,b) are used in the limit as x, t ! 0, whereas the notion of absolute stability applies at a nite mesh spacing ( x t). Example 4.1.1. In Chapter 2, we showed that the forward time-centered space scheme (4.1.2) is absolutely stable in L1 and L2 for r 1=2 when homogeneous boundary conditions (f (t) = g(t) = 0) are prescribed. 4.1. Implicit Di erence Methods 11 De nition 4.1.2. Let the di erence scheme (4.1.22a) depend on a certain minimal num- ber of parameters involving the mesh spacing. Its region of absolute stability is the region of parameter space where it is absolutely stable. De nition 4.1.3. A nite di erence scheme (4.1.22a) is unconditionally stable if it is absolutely stable for all choices of mesh spacing x and t. Example 4.1.2. The weighted average scheme (4.1.7b) depends on two parameters r and . We have shown that (4.1.7b) is unconditionally stable in L2 for a periodic initial value problem when 2 1=2 1]. It is absolutely stable in the regions f 2 1=2 1] r > 0g and f 2 0 1=2) r 1=2(1 ; 2 )g. Let us conclude our discussion of the weighted average scheme (4.1.7b) by examining the local discretization error n+1 n 2 un+1 + (1 ; ) 2 un j j n (u ; u )n ; uj ; uj ; : (4.1.23a) t xx j j 2 t x Using a Taylor's series expansion about (j x n t), we nd 1 x2 n n n 2 4 (4.1.23b) j = ; t( 2 ; )(utt )j + 12 (uxxxx)j + O( t ) + O( x ): Thus, the local discretization error is O( t)+O( x2 ), unless = 1=2, where the accuracy is O( t2)+ O( x2). The higher-order accuracy of the Crank-Nicolson scheme relative to other choices of is due to the centering of the nite di erence scheme about (j n + 1=2) on a uniform mesh. Problems 1. The local discretization error of the explicit forward time-centered space scheme (4.1.2) satis es (cf. (3.1.2)) x2 tn n 2 4 n j = ; 2 (utt )j + 12 (uxxxx)j + O( t ) + O( x ): Thus, the local discretization error is O( t) + O( x2) for arbitrary values of r. Is there a particular choice of r for which the discretization error is of higher order? 2. Use (4.1.23b) and the heat equation (4.1.1a) to show that the value = 1=2 ; x2 =12 t gives a weighted average scheme with an O( t2) + O( x4) local discretization error. Show that the absolute stability restriction (4.1.20) is satis ed for this choice of . 12 Parabolic PDEs 4.2 Neumann Boundary Conditions Thus far, we have considered problems with either periodic or Dirichlet boundary conditions. Let us now investigate problems having Neumann boundary conditions, e.g., ut = uxx 0<x<1 u(x 0) = (x) t>0 0x1 u(0 t) = f (t) ux(1 t) = g(t) (4.2.1a) (4.2.1b)...
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