45 js j s the coe cients cs jsj s can depend on j n

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Unformatted text preview: (j n) gives 22 X ; t jn = un + t(ut )n + O( t2 ) ; cs un + s x(ux)n + s 2 x (uxx)n + O( x3 )] ; tfjn : j j j j j jsj S Let us use the di erential equation (4.4.1) to eliminate ut in favor of spatial derivatives to obtain ; t jn = un + t jn (uxx)n + an(ux)n + bnun + fjn] + O( t2) j j j j jj ; X jsj Regrouping terms 22 cs un + s x(ux)n + s 2 x (uxx)n + O( x3)] ; tfjn: j j j S ; t jn = 1 ; ; x2 Enforcing that n j X jsj S cs + tbn ]un + ; x jj X jsj S scs + tan ](ux)n+ j j X s2 cs + t jn ](uxx)n + O( t2) + O( x3): j jsj S 2 ! 0 as x t ! 0 yields the consistency conditions X lim!0 1t cs ; 1] = bn j xt jsj S (4.4.7a) X n lim!0 x xt t jsj S scs = aj (4.4.7b) 34 Parabolic PDEs 2 X s2 cs n lim!0 xt xt 2 = j: jsj S (4.4.7c) Additional terms in the Taylor's series expansion may give rise to other consistency conditions if, e.g., mesh ratios such as x3 = t did not approach zero. However, as we have seen, it's usually necessary to require x2 = t to be bounded for an explicit scheme. Thus, one would hardly compute with the smaller t = O( x3). Stability is usually governed by the highest order terms in the di erence equation. Let us demonstrate this by performing a von Neumann analysis of the explicit scheme (4.4.5) assuming that coe cients are constant (cs, jsj S , are independent of j and n), fjn = 0, and computation is performed with constant t= x2 . Under these restrictions, the von Neumann analysis produces the ampli cation factor Xk Mk = cs!s !s = e2 is=J : (4.4.8) jsj S Suppose the coe cients cs, jsj powers of t1=2 of the form S , are smooth enough to have series expansions in cs = c0 + t1=2 c1=2 + tc1 + : : : : s s s (4.4.9) Substituting (4.4.9) into the consistency conditions (4.4.7) yields an alternate set of consistency conditions having the form X0 X 1=2 X1 cs = 1 cs = 0 cs = b (4.4.10a) jsj S X0 scs = 0 jsj S jsj S jsj S r x2 X sc1=2 = a t jsj S s x2 X s2c0 = : s t jsj S 2 (4.4.10b) (4.4.10c) The subscript j and superscript n on the coe cients a, b, and have been omitted for this constant coe cient analysis. Remark 1. We encountered the rst consistency condition in (4.4.10) in conjunction with the Maximum Principle (Theorem 3.1.1). 4.4. Variable-Coe cient and Nonlinear Problems 35 Substituting the expansions (4.4.9) into (4.4.8) yields an expansion of the ampli cation factor as Mk = Mk0 + t1=2 Mk1=2 + tMk1 + : : : (4.4.11a) where Mkl = X jsj S k cls!s l = 0 1 ::: : (4.4.11b) Using (4.4.10), we see that the leading term Mk0 is independent of a and b. Thus, Mk0 is the ampli cation factor of the di erential equation ut = uxx: Suppose that the principal part of the operator is stable, i.e., there exists a constant m0 k such that jMk0j 1 + m0 t: k k For simplicity, let us assume that S is small relative to J and expand !s in a series. Using (4.4.10) and (4.4.11b), this gives t Mk0 = 1 ; x2 ( 2J k )2 + O(( 2 JkS )3) r t ( 2 k ) + O(( 2 kS )3): x2 J J An expansion of Mk1 is not be needed. These equations imply that Mk0 is real and Mk1=2 is imaginary. Assuming this to be the case and using (4.4.11a) gives M 1=2 k = ia jMk j = (jMk0 j2 + tjMk1=2 j2 + : : : )1=2 : Expanding the square root jMk j 1 + O( t): Thus, the di erence scheme (4.4.5) satis es the von Neumann condition and is stable according to Theorem 3.2.1. The proof of stability when Mk0 and Mk1 are complex follows similar lines 17]. 36 Parabolic PDEs The previous discussion quantify the e ects of lower-order terms on stability when the coe cients are constant. Richtmyer and Morton 17], Chapter 5, show that a constant coe cient stability analysis with local values of the coe cients is su cient for the stability of a variable coe cient problem when (i) (4.4.5) is consistent and (ii) the coe cients cs, jsj S , are continuous and uniformly bounded on the domain of the di erential equation as t x ! 0. The local constant-coe cient analysis requires that stability be checked for all values of the coe cients in the domain. Consider, for example, solving a heat conduction problem with a variable di usivity (x) for x 2 (0 1). Using the explicit forward time-centered space scheme, we would have to show that ) 1 r = (xx2 t 2 8x 2 (0 1): 4.4.2 Convection-Di usion Problems The prior arguments imply that the lower-order terms in a parabolic di erential equations do not appreciably a ect stability however, caution is needed when absolute stability is required. Consider the convection-di usion equation ut = uxx + aux (4.4.12) which arises throughout uid mechanics. The equation clearly has properties of both the kinematic wave equation (convection) and the heat equation (di usion) and we would expect it to behave more like one or the other depending on the relative sizes of a and . Let's discretize this equation by the forward time-centered space di erence approximation to obtain Ujn+1 = Ujn + r(Ujn;1 ; 2Ujn + Ujn+1) + 2 (Ujn+1 ; Ujn;1) (4.4.13) where is the Courant number. It is not hard to show that solutions of initial value problems for (4.4.12) with periodic or compact data do not grow in time. Thus, it is reasonable to require kUnk kU0k: Stability analyses involving only the hig...
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