# 11 can be determined by diagonalizing the jacobian

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Unformatted text preview: Jacobian (10.1.2b). This can be done for hyperbolic systems since A(u) has m distinct eigenvalues (De nition 10.1.1). Thus, let P = p(1) p(2) : : : p(m) ] (10.1.6a) and recall the eigenvalue-eigenvector relation AP = P (10.1.6b) 4 Hyperbolic Problems where 2 6 =6 6 4 3 7 7 7 5 1 2 ... (10.1.6c) m Multiplying (10.1.2a) by P;1 and using (10.1.6b) gives P;1ut + P;1Aux = P;1ut + P;1ux = P;1b: Let w = P;1u so that (10.1.7) wt + wx = P;1ut + (P;1)tu + P;1ux + (P;1)xu]: Using (10.1.7) wt + wx = Qw + g (10.1.8a) where Q = (P;1)t + (P;1)x]P g = P;1b: (10.1.8b) i = 1 2 : : : m: (10.1.8c) In component form, (10.1.8a) is (wi)t + i(wi)x = m X j =1 qi j wj + gi Thus, the transformation (10.1.7) has uncoupled the di erentiated terms of the original system (10.1.2a). Consider the directional derivative of each component wi, i = 1 2 : : : m, of w, dwi = (w ) + (w ) dx i = 1 2 ::: m it ix dt dt in the directions dx = i = 1 2 ::: m (10.1.9a) dt i and use (10.1.8c) to obtain m dwi = X q w + g i = 1 2 : : : m: (10.1.9b) dt j=1 i j j i 10.1. Conservation Laws 5 The curves (10.1.9a) are called the characteristics of the system (10.1.1, 10.1.2). The partial di erential equations (10.1.2) may be solved by integrating the 2m ordinary differential equations (10.1.9a, 10.1.9b). This system is uncoupled through its di erentiated terms but coupled through Q and g. This method of solution is, quite naturally, called the method of characteristics. While we could develop numerical methods based on the method of characteristics, they are generally not e cient when m > 2. De nition 10.1.2. The set of all points that determine the solution at a point P (x0 t0) is called the domain of dependence of P . Consider the arbitrary point P (x0 t0 ) and the characteristics passing through it as shown in Figure 10.1.2. The solution u(x0 t0 ) depends on the initial data on the interval A B ] and on the values of b in the region APB , bounded by A B ] and the characteristic curves x = 1 and x = m . Thus, the region APB is the domain of dependence of P . _ _ t P(x 0 ,t 0) 11...
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