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Unformatted text preview: on. 10 Hyperbolic Problems x = x 0 + at t 1 a x x0 at u(x,t) u(x,t) = φ(x-at) u(x,0) = φ(x) x at Figure 10.1.5: Characteristic curves and solution of the initial value problem (10.1.11a, 10.1.13) when a is a constant. Example 10.1.4. The simplest case occurs when a is a constant and f (u) = au. All of the characteristics are parallel straight lines with slope 1=a. The solution of the initial value problem (10.1.11a, 10.1.13) is u(x t) = u0(x ; at) and is, as shown in Figure 10.1.5, a wave that maintains its shape and travels with speed a. Example 10.1.5. Setting a(u) = u and f (u) = u2 =2 in (10.1.11a, 10.1.11b) yields the inviscid Burgers' equation 1 (10.1.16) ut + 2 (u2)x = 0: Again, consider an initial value problem having the initial condition (10.1.13), so the characteristic is given by (10.1.15) with a0 = u(x0 0) = u0(x0 ), i.e., x = x0 + u0(x0 )t: (10.1.17) The characteristics are straight lines with a slope that depends on the value of the initial data thus, the characteristic passing through the point (x0 0) has slope 1=u0(x0 ). 10.1. Conservation Laws 11 The fact that the characteristics are not parallel introduces a di culty that was not present in the linear problem of Example 10.1.4. Consider characteristics passing through (x0 0) and (x1 0) and suppose that u0(x0 ) > u0(x1 ) for x1 > x0. Since the slope of the characteristic passing through (x0 0) is less than the slope of the one passing through (x1 0), the two characteristics will intersect at a point, say, P as shown in Figure 10.1.6. The solution would appear to be multivalued at points such as P . t P x = x 0 + φ (x 0 )t 1 1 φ1 φ0 x0 x = x 1 + φ (x 1 )t x x1 Figure 10.1.6: Characteristic curves for two initial points x0 and x1 for Burgers' equation (10.1.16). The characteristics intersect at a point P . In order to clarify matters, let's examine the speci c choice of u0 given by Lax 20] 8 if x < 0 <1 u0(x) = : 1 ; x if 0 x < 1 : (10.1.18) 0 if 1 x Using (10.1.17), we see that the characteristic passing through the point (x0 0) s...
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