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 eigenvalueeigenvector 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 .
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_
t P(x 0 ,t 0) 11...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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