MIT OpenCourseWarehttp://ocw.mit.edu 18.306 Advanced Partial Differential Equations with Applications Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Problem Set Number 03 18.306 — MIT (Fall 2009) Rodolfo R. Rosales MIT, Math. Dept., Cambridge, MA 02139 October 23, 2009 Due: Friday October 30. Contents 1.1 Statement: Linear 1st order PDE (problem 10) T.L.R. . . . . . . . . . . . . . . . . 1 1.2 Statement: Riemann Problems (problem 01) . . . . . . . . . . . . . . . . . . . . . . 2 Riemann Problem for ut + (0.5 u 2)x = 1. . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Justification of quadratic fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Statement: Riemann Problems (problem 02). . . . . . . . . . . . . . . . . . . . . . 3 Riemann Problem for ut + (0.5 u 2)x = δ(x). . . . . . . . . . . . . . . . . . . 3 1.3.1 What the equation means. Causality. . . . . . . . . . . . . . . . . . . . . . 4 Rankine Hugoniot conditions for shocks with point sources. . . . . . . . . . . . . 8 1.4 Statement: KdV-Burgers Equation (problem 01). . . . . . . . . . . . . . . . . . . . 9 List of Figures 1.1 Cases P1 – P3: typical characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Cases P4 – P5 and Ex: typical characteristics. . . . . . . . . . . . . . . . . . . . . . 5 1.1 Statement: Linear 1st order PDE (problem 10) T.L.R. Do the Task Left to the Reader (T.L.R.) at the end of the answer to the problem Linear 1st order PDE (problem 10) in Problem set # 2 (this is in p.p. 14-15 of the Answers to Probelm Set Number 1� � � � 02 — posted in the web page). 1.2 Statement: Riemann Problems (problem 01). Consider the following conservation law (in a-dimensional variables) 1 2 ut + u = 1, for − ∞ < x < ∞ and t > 0, (1.1)2 x where u is conserved, and shocks are used to avoid multiple-valued solutions. Find the solution to the Riemann problem for this equation. Namely, for the initial values u(x, 0) = a for x < 0 and u(x, 0) = b for x > 0, (1.2) where a and b are arbitrary real constants −∞ < a, b < ∞. Hint: The solution involves shocks, expansion fans, and regions where u depends on time only. Expansion fans are regions where all the characteristics emanate from a single point in space time. 1.2.1 Justification of quadratic fluxes. Here we justify the use of conservation laws of the form 1 2 ut + u = S, for − ∞ < x < ∞ and t > 0, (1.3)2 x where S is some source term, u is conserved and can be both positive or negative, and shocks are used to avoid multiple-values in the solution. Consider a scalar conservation law in 1-D, with a source term, ρt + qx = S, (1.4) where ρ = ρ(x, t) is the density of some conserved quantity (hence ρ ≥ 0), q = Q(ρ) is the corre-sponding flux, and S = S(ρ, x) is the density of sources/sinks. Assume now that S is “small” — this is made precise below in item 2. Then solutions where ρ is close to a constant should be possible. Hence let ρ0 > 0 be some fixed (constant) density value, and proceed as follows: 2� � � � 1. Expand Q near ρ0 using Taylor’s theorem Q = q0 + c0 (ρ − ρ0) + s1 (ρ − ρ0)2 + . . . , (1.5)2 ρ0 where q0 is the flux for ρ = ρ0, c0 is the corresponding characteristic speed, and s1 is a constant with the dimensions of a velocity. We now assume that s1 �= 0; in fact, that1 s1 > 0. Notice that s1 is a measure of how nonlinear the equation in (1.4) is. The further away for zero s1 is, the stronger the leading order nonlinear term in the equation is. 2. Let S1 > 0 be some “typical” value for the source term size, and let L > 0 be some “typical” L S1 length scale. Then the source term is small in the sense that 0 � �2 = � 1. s1 ρ0 Introduce the a-dimensional variables ˜˜x˜ = x − c0 t and t = � s1 t, with ρ = ρ0 (1 + � u) and S = S1 S. (1.6)L L Then (1.4) becomes 1 2 ˜ut˜+ u + O(�) = S. (1.7)2 x˜Upon neglecting the O(�) term, this has the form in (1.3). 1.3 Statement: Riemann Problems (problem 02). Consider the following conservation law (in a-dimensional variables) for the density u 1 ut + u 2 = δ(x), for − ∞ < x < ∞ and t > 0, (1.8)2 x where shocks are used to avoid multiple-valued solutions, and δ(∗) stands for Dirac’s delta function. Solve the Riemann problem for (1.8) given by the initial data u(x, 0) = a for x < 0 and u(x, 0) = b for x > 0, (1.9) where −∞ < a, b < ∞ are arbitrary real constants. Hint: The solution involves shocks, expansion fans,2 and regions where u is constant. The information in § 1.3.1 should prove useful. 1If s1 < 0, a similar analysis is possible. 2Expansion fans are regions where all the characteristics emanate from a single point in space time. 3� � � � � � 1.3.1 What the equation means. Causality. The delta function (point source term) has meaning via the integral form of the conservation law; namely: d xR 1 1 u dx = 1 + u 2(xL, t) − u 2(xR, t), (1.10)dt xL 2 2 for any constants xL < 0 < xR. Hence the solutions to (1.8) are “regular” solutions of the conser-vation law 1 2 ut + u = 0, (1.11)2 x away from the position x = 0 of the point source, and have a discontinuity at x = 0 satisfying uR 2 − uL 2 = 2, (1.12) where uL (resp. uR) is the value of the solution on the left (resp. right) side of the discontinuity. In addition, the discontinuity at x = 0 should satisfy causality: Every point in space time should be connected, via a single (1.13) characteristic, to a point in the past where data …