Conservation laws - II (Weak solutions)

In this article, we will mainly discuss some standard numerical techniques to analyse two types of Hyperbolic partial differential equations:

Advection equation

\begin{equation} u_t + a(x,t)~u_x = 0 \label{advection} \end{equation}

scalar conservation laws

\begin{equation} u_t + (f(x,t,u))_x = 0 \label{conservation} \end{equation}

The conservation law equation (\ref{conservation}) models the evolution of a conservative quantity like mass, momentum, and energy. That is a quantity whose variation inside a closed domain is equal to its flux across the boundaries, i.e, the amount which flows in minus the amount which flows out, of the closed domain.

The advection and the conservation law equations are intimately interconnected. In the case when \(f = f(u)\), the conservation equation Eq. 2 can be rewritten in the advective form as

\begin{equation} u_t + \dfrac{df}{du}u_x = 0 \end{equation}

and when a is constant with respect to x, the advection equation can also be viewed as a conservation law. For example, a very commonly used prototype example of a nonlinear conservation law is the celebrated Burger’s equation

\begin{equation} u_t + \dfrac{1}{2} (u^2)_x = 0 \label{burgers} \end{equation}

which becomes

\begin{equation} u_t + u~u_x = 0 \end{equation}

when written in advective form.

The method of characteristics

Consider the quasi-linear equation of the form

\begin{equation} u_t + a(x,t,u)~u_x + b(x,t,u) = 0 \label{quasi} \end{equation}

Let \(x = x(t)\) be a parametric curve in the (\(x, t\)) plane such that \(\dot{x}= a(x, t, u(x(t), t))\), where \(u(x(t), t) = z(t)\) is the solution to Eq. \ref{quasi} along this curve. Using the chain rule and plugging into Eq. \ref{quasi} yields

\begin{equation} \dot{z} = \dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x}~\dot{x} = \dfrac{\partial u}{\partial t} + a(x,t,u)~\dfrac{\partial u}{\partial x} = -b(x,t,u) = -b(x,t,z) \end{equation}

i.e, finding a solution for the quasi-linear equation reduces to solving the following system of first order ordinary differential equations.

\begin{equation} \begin{aligned} \dot{x} &= a(x,t,z) \\ \dot{z} &= -b(x,t,z) \\ x(0) &= x_0 \quad z(0) = u(x_0, 0) = u_0(x_0) \end{aligned} \label{quasicharact} \end{equation}

Equations \ref{quasicharact} are known as the characteristic equations and the resulting solution curves \(x = x(t)\), \(x(0) = x_0\) are called characteristic curves.

For example, The characteristic equations for the advection equation is

\begin{equation} \begin{aligned} \dot{x} &= a \\ \dot{z} &= 0 \end{aligned} \end{equation}

whose solution is \(x(t) = x_0 + at\), \(z(t) = u(x(t), t) = u_0(x_0)\). Two key important points should be noted here. i) The characteristic curves are straight lines. ii) The solution u is constant along the characteristic lines. Note that when \(a > 0\) the characteristic lines are directed to the right and when \(a < 0\) they are directed to the left. In some sense the sign of \(a\) indicates the direction of propagation of information. In fact, the advection equation is also called the one-way wave equation, where \(a\) is the speed of propagation of the wave.

To find the solution \(u(x, t)\) at an arbitrary point (\(x, t\)) in the x-t plane, one needs to follow the characteristic line passing through (\(x, t\)) back to its original point at \(t\) equal to 0: (\(x_0 , 0\)) with \(x = x_0 + at\). This leads to

\begin{equation} u(x,t) = u_0(x_0) = u_0(x-at) \end{equation}

The system of characteristic equations for Burger’s equation is given by

\begin{equation} \begin{aligned} \dot{x} &= z \\ \dot{z} &= 0 \end{aligned} \end{equation}

The characteristic solution is thus given by

\begin{equation} u(x(t), t) = u_0(x_0) \end{equation}

where \(x(t) = x_0 + u_0(x_0)t\).

Again note that the characteristics are straight lines and the solution is constant along the char-acteristic lines, with one important difference, however; the characteristic curves are no longer parallel to each other. As we will see below this has rather unpleasant consequences. Provided the characteristics lines do not cross each other, which is guaranteed for at least a short period of time if the initial data \(u_0\) is continuous, the solution to Burgers equation is given by the following implicit formula

\begin{equation} \begin{aligned} u(x,t) &= u_0(x-u_0(x_0)~t) \\ x &= x_0 + u_0(x_0)t \end{aligned} \end{equation}

We would notice that, the slope of the characteristic curves \(x = x_0 + u_0(x_0)t\) for Burger’s equation (Eq. \ref{burgers}) increases when \(u_0(x_0)\) increases and decreases when \(u_0(x_0)\) decreases. Therefore, the characteristic curves will accordingly diverge or converge toward each other. The aforementioned behaviour can cause two convergent characteristic lines to ultimately cross each other at some point in the x-t plane. Beyond such intersection point the characteristic solution is no-longer valid, because the value of \(u(x, t)\) at such a point is not univalued–one can follow back either one of the two intersecting characteristic lines.

One way to correct for this flaw is by stopping the characteristic lines as soon as they cross each other. Let \(\Sigma\) be the set of such crossing points in the x-t plane. The solution can then be defined on both sides of \(\Sigma\) by following the corresponding characteristic line back to its origin. Below we will see that \(\Sigma\) is a parametric curve on the form \(x = s(t)\) and constitutes a curve of discontinuity for \(u(x, t)\). Such a curve is called a shock by analogy to gas dynamics. One of the main difficulties in practice is to find the shock curve \(x = s(t)\). For any given (\(x_1, t_1\)), one has to determine whether two characteristic lines cross each other prior to time \(t_1\) along the curve \(x = x_1\).

After a shock is formed the solution \(u(x, t)\) is no longer valid in the classical sense except for its restrictions on the sub-domains located on either side on the shock. Nevertheless, such solution can be defined in the weak sense on the whole x-t plane.

Definition (Weak solution) A function \(u(x, t)\) is said to be a weak solution for the conservation law \(u_t + (f(x, t, u))_x = 0\), if for any test function \(\phi(x,t)\) sufficiently smooth with a compact support in (\(- \infty, + \infty\) \() \times\) \((0, + \infty)\), the solution \(u(x, t)\) satisfies
\begin{equation} \int \limits_0^{+ \infty} \int \limits_{-\infty}^{+\infty} u(x,t)~\phi_t(x,t) dxdt + f(x,t,u)~\phi_x(x,t) dxdt = 0. \end{equation}

The notion of weak solutions is more general and the set of weak solutions contains discontinuous solutions as well as the classical \(\mathit{C}^1\) solutions as a special subset. However, in some situations weak solutions are not unique, in the sense that one initial value problem can have more than one weak solution. Selecting the physically relevant solution can be tricky. However, common sense suggests that for any given Cauchy problem, only one solution is physically relevant. We need an additional constraint to choose this physically relevant solution among all the weak solutions. In fact, in reality some viscosity is always associated with a given conservation law, so instead we have

\begin{equation} u_t^{\epsilon} + (f(u^{\epsilon}))_x = \epsilon~\dfrac{\partial^2 u^{\epsilon}}{\partial x^2} \label{viscous} \end{equation}

and the zero viscosity limit, \(\epsilon \rightarrow 0\), is just a convenient mathematical idealization. The above mentioned point is true in many physical applications. Therefore, one universally accepted criterion states that the physically relevant weak solution for the conservation law \(u_t + (f(u))_x = 0\) is the limit of the solution to the viscous equation (Eq. \ref{viscous}) when \(\epsilon \rightarrow 0\). On the other hand, it is easy to show that, when combined with appropriate initial conditions, the latter has a unique solution. In practice, however, it is not clear how to establish if a given weak solution is actually the vanishing viscosity limit or not. The answer to this question is provided by the concept of entropic solutions. In a nutshell, the entropy condition states that the physical solution satisfies a general principle of thermodynamics that the entropy always decreases. It remains to find which among the weak solutions for a given conservation law satisfies the so-called entropy condition.

The theorem below provides the necessary ingredients both for constructing weak solutions and to gain physical insight.

Theorem (Rankine-Hugoniot Condition) Let \(\Sigma\) be a curve in (\(- \infty, + \infty\)) \(\times\) \((0, + \infty)\) parametrized by \(x = s(t)\). Let \(u(x, t)\) be a \(\mathit{C}^1\) function on both side of but possibly not defined and discontinuous across the curve \(\Sigma\). Assume \(u\) is a solution to the conservation law \(u_t + (f(u))_x = 0\) on all points \((x, t)\) not on \(\Sigma\). For each point \((x_1,t_1) \in \Sigma\) we set

\begin{equation} u(x_1,t_1) = \lim \limits_{\Omega_1 \cup \Omega_2 \cup \ni (x,t) \rightarrow (x_1,t_1)} u(x,t) \end{equation}

i.e. the limits from the right and from the left of \(\Sigma\). Then \(u(x, t)\) is a weak solution for the conservation law if and only if the shock speed \(\dot{s}\) satisfies

\begin{equation} \dot{s} = \dfrac{ds}{dt} = \dfrac{f(u_{+}) - f(u_{-})}{u_{+} - u_{-}} \end{equation}