Conservation Laws - Mathematical Method

The Method of Characteristics¶

Integral Surface¶

Consider the general quasilinear first order PDE:

a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)

(1)

We note that the vector field $(-u_x, -u_y, 1)^T$ would be orthogonal to the solution $u(x,y)$ everywhere. Therefore the vector field $(a, b, c)^T$ is perpendicular to $(-u_x, -u_y, 1)^T$ thus must lie in the tangent plane of $u(x,y)$ . Surfaces that are tangent to a vector space is called integral surfaces, hence to find the solution we want to find the integral surfaces of $u(x,y)$ .

We do this by:

Method of Characterisitcs

Parametrize the inital condition $\Gamma(r)$
Find integral curves $\chi$ starting from one instance of the initial condition(at some $r$ ) by solving a system of ODE. Concretely, suppose $\chi(s) = (x(s),y(s),z(s))$ is a parametrization of the integral curves, then it must satisfy that:
$\begin{cases} \frac{dx}{ds}&=a(x,y,u) \\ \frac{dy}{ds}&=b(x,y,u) \\ \frac{dz}{ds}&=c(x,y,z) \\ \end{cases}$
(2)
on some small $|t-t_0|<\delta$ . The resulting integral curves are functions of $r,s$ where $r$ specifies where the evolution start and $s$ specifies how much the integral curve grow away from the $\Gamma$ .
Compute the integral surface as a continuous union by replacing $r,s$ with $x,y$

Example 1 (Constant Wave-speed Advection)

Consider the equation $u_t+cu_x=0$ with the initial condition $u(t,0)=\exp(-x^2)$ . We parametrize the initial condition as $(t,x,z)=(0,r,\exp(-r^2))$ . Fix $r$ , the system is:

\begin{cases} \frac{dt}{ds}&=1 \\ \frac{dx}{ds}&=c \\ \frac{dz}{ds}&=0 \\ \end{cases} \Longrightarrow \begin{cases} t(r,s)=s+C(r) \\ x(r,s)=cs+D(r)\\ z(r,s)=E(r) \\ \end{cases}

(3)

Substituting the initial condition, we get:

\begin{cases} t(r,s)=s \\ x(r,s)=cs+r\\ z(r,s)=\exp(-r^2) \\ \end{cases}

(4)

But now $r=x-cs=x-ct$ hence we get the solution:

u(t,x)=\exp(-(x-ct)^2)

(5)

Visualizing the solution:

We showed the special case when $c=1$ , the lines are the integral curves above. In this special case the value of $z=u(x,y)$ is fixed on the curve.

The Finite Volume Method¶

Hyperbolic Conservation Law¶

For simplicity, we consider the homogenous case:

u_t + a(t,x)u_x = 0

(6)

If we discretize the spatial domain into an array of contiguous cells called control volumes, we can see that the equation encapsulate a sense of conservation. Suppose $[a,b]$ is a cell of interest, integrate the equation by parts results in :

\frac{d}{dt}\int_{a}^b u dx = \underbrace{a(t,a)u(t,a)}_{\text{flux in}}-\underbrace{a(t,b)u(t,b)}_{\text{flux out}} - \underbrace{\int_a^b a_x(t,x)udx}_{\text{source/disspation}}

(7)

In particular, if we have $a(t,x)=a(t)$ , then we have no source and disspation. This motivates us to define:

For a simple strict conservation law we have:

\frac{d}{dt}\int_a^b u = \underbrace{f(u(a,t))}_{\text{flux in}}-\underbrace{f(u(b,t))}_{\text{flux out}}

(10)

It is clear that:

The idea behind finite volume¶

Finite volume method utillise the conservation structure of the equation. Specifically, we enforce that conservation law holds at each cell $[x_{i-1/2},x_{i+1/2}]$

\int_{t_j}^{t_{j+1}}\left\{ \int_{x_{i-1/2}}^{x_{i+1/2}} \left[ u_t + (f(u))_x -g\right]dx \right\}dt=0

(11)