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Unformatted text preview: le 10.1.8. A Riemann problem is an initial value (Cauchy) problem for (10.1.1) with piecewise-constant initial data. Riemann problems play an important role in the numerical solution of conservation laws using both nite di erence and nite element techniques. In this introductory section, let us illustrate a Riemann problem for the inviscid Burgers' equation (10.1.16). Thus, we apply the initial data u(x 0) = uL iiff x < 0 : (10.1.25) uR x 0 As in the previous two examples, we have to distinguish between two cases when uL > uR and uL uR . The solution may be obtained by considering piecewise-linear continuous initial conditions as in Examples 10.1.6 and 10.1.7, but with the \ramp" extending from 0 to instead of from 0 to 1. We could then take a limit as ! 0. The details are left to an exercise (Problem 1 at the end of this section). When uL > uR , the characteristics emanating from points x0 < 0 are the straight lines x = x0 + uLt (cf. (10.1.17)). Those emanating from points x0 > 0 are x = x0 + uR t. The characteristics cross immediately and a shock forms. Using (10.1.20), we see that the shock moves with speed _ = (uL + uR)=2. The solution is constant along the characteristics and, hence, is given by u(x t) = uL iif x=t < (uL + uR)=2 uL > uR: (10.1.26a) uR f x=t (uL + uR)=2 16 Hyperbolic Problems t 1 0 x 1 Figure 10.1.11: Characteristics for Example 10.1.7. u(x,0) u(x,1/2) 1 1 1 0 2 0 x u(x,1) 1 2 x 2 x u(x,3/2) 1 1 0 1 2 x 0 1 Figure 10.1.12: Solution u(x t) of Example 10.1.7 as a function of x at t = 0, 1/2, 1, and 3/2. 10.1. Conservation Laws 17 Several characteristics and the location of the shock are shown in Figure 10.1.13. When uL uR, the characteristics do not intersect. There is a region between the characteristic x = uLt emanating from x0 = 0; and x = uRt emanating from x0 = 0+ where the initial conditions fail to determine the solution. As determined by either the limiting process suggested in Problem 1 or thermodynamic arguments using entropy considerations 20], no sho...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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