Method of Characteristics

Consider the scalar conservation law,

Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \]

Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.

Use the method of characteristics to solve

with

Consider the scalar conservation law,

Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1-x, & 0<x<1. \end {cases} \]

Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed \(u(x,y,t)\) flowing down a wall, see Fig. 1. The paint is assumed to be uniform in the \(z\)-direction. The balance between gravity and viscosity (fluid friction) means that the velocity satisfies the equation \[ \frac {\partial ^2u}{\partial y^2}=-c, \] where \(c\) is a positive constant. This is supplemented by the boundary conditions \[ u(x,0,t)=0,\qquad \left .\frac {\partial u(x,y,t)}{\partial y}\right |_{y=h}=0. \] The density of paint per unit length in the \(x\)-direction is \(\rho _0h(x,t)\) where \(\rho _0\) is a constant, and the corresponding flux is \[ q(x,t)=\rho _0\int _0^h u(x,y,t)d y. \]

1. Using conservation of paint, and solving for \(u(x,y,t)\) in terms of \(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\): \[ \frac {\partial h}{\partial t}+ch^2\frac {\partial h}{\partial x}=0. \]
2. Set \(c=1\). Show that the characteristics are straight lines and that the Rankine-Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {[h^3/3]^+_-}{[h]^+_-}. \]
3. A stripe of paint is applied at \(t=0\) so that \[ h(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \] Show that, for small enough \(t\), \[ h= \begin {cases} 0, & x<0,\\ (x/t)^{1/2}, & 0<x<t,\\ 1, & t<x<S(t),\\ 0, & S(t)<x, \end {cases} \] where the shock is \(x=S(t)=1+t/3\).
4. Explain why this solution changes at \(t=3/2\), and show that thereafter \[ \frac {dS}{dt}=\frac {S}{3t}. \]

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed \(u(x,y,t)\) flowing down a wall. The paint is assumed to be uniform in the \(z\)-direction. The balance between gravity and viscosity (fluid friction) means that the velocity satisfies the equation \[ \frac {\partial ^2u}{\partial y^2}=-c, \] where \(c\) is a positive constant. This is supplemented by the boundary conditions \[ u(x,0,t)=0,\qquad \left .\frac {\partial u(x,y,t)}{\partial y}\right |_{y=h}=0. \] The density of paint per unit length in the \(x\)-direction is \(\rho _0h(x,t)\) where \(\rho _0\) is a constant, and the corresponding flux is \[ q(x,t)=\rho _0\int _0^h u(x,y,t)d y. \]

1. Sketch a figure illustrating the physical problem.
2. Using conservation of paint, and solving for \(u(x,y,t)\) in terms of \(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\): \[ \frac {\partial h}{\partial t}+ch^2\frac {\partial h}{\partial x}=0. \]
3. Set \(c=1\). Show that the characteristics are straight lines and that the Rankine-Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {[h^3/3]^+_-}{[h]^+_-}. \]
4. A stripe of paint is applied at \(t=0\) so that \[ h(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \] Show that, for small enough \(t\), \[ h= \begin {cases} 0, & x<0,\\ (x/t)^{1/2}, & 0<x<t,\\ 1, & t<x<S(t),\\ 0, & S(t)<x, \end {cases} \] where the shock is \(x=S(t)=1+t/3\). Explain why this solution changes at \(t=3/2\), and show that thereafter \[ \frac {dS}{dt}=\frac {S}{3t}. \]

Suppose that \(\rho (x,t)\) is the number density of cars per unit length along a road, \(x\) being distance along the road, such that \[ \frac {\partial \rho }{\partial t} +\frac {\partial [\rho (1-\rho )]}{\partial x}=0. \]

1. Show that \(\rho \) is constant along the characteristics \[ \frac {dx}{dt}=1-2\rho , \] and derive the following Rankine-Hugoniot condition for the speed of a shock \(x=S(t)\): \[ \frac {dS}{dt}=\frac {[\rho (1-\rho )]_-^+}{[\rho ]_-^+}. \]
2. A queue is building up at a traffic light \(x=1\) so that, when the light turns to green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & \text {if }x<0\text { and }x>1,\\ x, & \text {if }0<x<1. \end {cases} \] Solve the corresponding characteristic equations, and sketch the resulting characteristic curves. Deduce that a collision first occurs at \(x=1/2\) when \(t=1/2\), and that thereafter there is a shock such that \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

Consider the equation \[ \frac {\partial u}{\partial t} +\frac {\partial }{\partial x}\left (\frac {u^2}{2}\right ) =\epsilon \frac {\partial ^2u}{\partial x^2}, \] for \(-\infty <x<\infty \) and \(t>0\).

1. Look for a traveling wave solution \(u(x,t)=U(z)\), \(z=(x-Vt)/\epsilon \), with velocity \(V>0\) and \(U(z)\to U_\pm \) as \(z\to \pm \infty \). Solve the resulting ODE for \(U(z)\) and deduce that \[ V=\frac {[U^2/2]_{-\infty }^{\infty }}{[U]_{-\infty }^{\infty }} \]
2. Discuss how the traveling wave solution relates to shock solutions of the quasilinear equation obtained by setting \(\epsilon =0\).
3. Using phase-plane analysis, show that the wave profile \(U\) is monotonically decreasing, that is, \(dU/dz<0\).

Consider the one-dimensional parabolic PDE \[ \frac {\partial u}{\partial t} +\frac {\partial }{\partial x}\left (\frac {u^n}{n}\right ) =\epsilon \frac {\partial ^2u}{\partial x^2}. \]

1. Show that a traveling wave solution \(u(x,t)=U(z)\) with \(z=(x-Vt)/\epsilon \) satisfies \[ \frac {dU}{dz}=\frac {U^n}{n}-VU+\text {constant}, \] and deduce that \[ V=\frac {[U^n/n]_{-\infty }^{\infty }}{[U]_{-\infty }^{\infty }}. \]
2. Discuss how the traveling wave solution relates to shock solutions of the quasilinear equation obtained by setting \(\epsilon =0\). Also show that, when \(n=2\), \(U\) can only tend to \(U(\pm \infty )\) as \(z\to \pm \infty \) if \(dU/dz<0\).

Suppose that \(\rho (x,t)\) is the number density of cars evolving according to the traffic model \[ \frac {\partial \rho }{\partial t}+\frac {\partial (\rho u)}{\partial x}=0. \] with \(u\) the car speed. Take \(u=1-\rho \). A queue is building up at a traffic light \(x=1\) so that, when the light turns green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & x<0\text { and }x>1,\\ x, & 0<x<1. \end {cases} \]

1. Use the method of characteristics to solve the Cauchy problem. In particular, construct the characteristic projections and identify any shocks or rarefaction waves.
2. Hence, show that a collision first occurs at \(x=1/2\) when \(t=1/2\) and, thereafter, there is a shock \(S(t)\). Use the Rankine-Hugoniot condition to derive the following expression for the speed of the shock: \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

Suppose that \(\rho (x,t)\) is the number density of cars per unit length along a road, \(x\) being the distance along the road such that \[ \frac {\partial \rho }{\partial t} +\frac {\partial [\rho (1-\rho )]}{\partial x}=0. \]

1. Show that \(\rho \) is constant along characteristics \[ \frac {dx}{dt}=1-2\rho , \] and derive the following Rankine-Hugoniot conditions for the speed of a shock \(x=S(t)\): \[ \frac {dS}{dt}=\frac {[\rho (1-\rho )]_-^+}{[\rho ]_-^+}. \]
2. A queue is building up at traffic light \(x=1\) so that, when the light turns to green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & \text {if }x<0\text { and }x>1;\\ x, & \text {if }0<x<1. \end {cases} \] Solve the characteristic equations, and sketch the resulting characteristic curves. Deduce that a collision first occurs at \(x=1/2\) when \(t=1/2\), and that thereafter there is a shock such that \[ \frac {dS}{dt}=\frac {S+t+1}{2t}. \]

Suppose that \(\rho (x,t)\) is the number density of cars per unit length along a road, \(x\) being distance along the road, such that \[ \frac {\partial \rho }{\partial t} +\frac {\partial [\rho (1-\rho )]}{\partial x}=0. \]

1. Show that \(\rho \) is constant along the characteristics \[ \frac {dx}{dt}=1-2\rho , \] and derive the following Rankine-Hugoniot condition for the speed of a shock \(x=S(t)\): \[ \frac {dS}{dt}=\frac {[\rho (1-\rho )]_-^+}{[\rho ]_-^+}. \]
2. A queue is building up at a traffic light \(x=1\) so that, when the light turns to green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & \text {if }x<0\text { and }x>1,\\ x, & \text {if }0<x<1. \end {cases} \] Solve the corresponding characteristic equations, and sketch the resulting characteristic curves. Deduce that a collision first occurs at \(x=1/2\) when \(t=1/2\), and that thereafter there is a shock such that \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

Paint flowing down a wall has thickness \(u(x,t)\) satisfying (for \(t>0\)) \[ \frac {\partial u}{\partial t}+u^2\frac {\partial u}{\partial x}=0. \]

1. Show that the characteristics are straight lines and that the Rankine–Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {\left [u^3/3\right ]_-^+}{[u]_-^+}. \]
2. A stripe of paint is applied at \(t=0\) so that \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1\\ 1, & 0<x<1 \end {cases} \] For sufficiently small \(t\), determine \(u\) in the domains \(x<0\), \(0<x<t\), \(t<x<S(t)\) and \(S(t)<x\), where the shock is \(x=S(t)=1+t/3\)
3. Explain why the solution changes at \(t=3/2\) and show that thereafter \(\dot {S}=S/3t\).

Denote \(U=\{x\in \mathbb {R}^2:x_1>0\}\), and \(\Gamma =\{x\in \mathbb {R}^2:x_1=1\}\). Use the method of characteristics to solve the first-order problem \[ \begin {aligned} x_1u_{x_1}+uu_{x_2}-x_2 &= 0, && \text {in } U,\\ u(1,x_2) &= 2x_2, && \text {on } \Gamma . \end {aligned} \] (Hint: to solve the characteristic equations, note that the linear combinations \(x_2+z\) and \(x_2-z\) decouple.) Determine the region around \(\Gamma \) upon which the solution is well-defined.

Conservation Laws. Consider the first-order equation \[ (G(u))_x+u_y=0, \] where \(G\in C^\infty \), with initial condition \(u(x,0)=h(x)\).

1. Using the method of characteristics, derive the general solution of the initial value problem. Assuming \(h\in C^\infty \), find a condition on \(G'(h(x))\) which implies the solution \(u(x,y)\) will develop a singularity in finite time \(y>0\).
2. Set \(G(u)=\frac {1}{2}u^2\). Given \(u_0\in \mathbb {R}\), find explicit weak solution(s) for \[ h(x)= \begin {cases} u_0, & x<0,\\ 0, & x>0. \end {cases} \] Show that your solutions are weak solutions. Which properties do your solutions exhibit: shocks, rarefaction, nonuniqueness?

Using the method of characteristics, solve the initial value problem (3) assuming \(f(x,t)=x\) and \(\varphi (s)=s\).

Write a brief discussion of the method of characteristics and then demonstrate the method by solving \[ u_x+u_y=u^2;\qquad u(x,0)=x^2. \]

The quasilinear Cauchy problem \[ xu_x+yuu_y+xy=0,\qquad u=5\quad \text {when }xy=1, \] has the solution \[ u(x,y)=-1+\sqrt {38-2xy}. \]

1. Derive this solution by the method of characteristics. (Hint: Along a characteristic \(x(s)\), \(y(s)\), \(u(s)\) find \(d(xy)/ds\) and then \(du/dz\) where \(z=xy\).)
2. Verify that \(u(x,y)\) is a solution in the plane domain defined by \(xy<19\).