Conservation Laws

$$\mbox {Is a system of equations of the form} \left\{\begin{array}... ...0,{\bf x})={\bf u}_0(\bf x) & {\bf x}\in {\mathbb{R}}^n, t>0 \end{array}\right.$$ (134)

It's a conservation law since the flux ${\bf F}$ will be constant if $d {\bf u} /dt=0$. Conservation law equations are some of the most important and ubiquitous equations of Physics.

Conservation Laws have a connection to the wave equation. Take the case of one space dimension: if ${\bf f}$ is smooth enough, with $x\in {\mathbb{R}}^1$ for example:

$$\frac{\partial {\bf u}}{\partial t}+[f(\bf u)]_x=0\Rightarrow \fr... ...ial t}+\frac{\partial f}{\partial {\bf u}}\frac{\partial {\bf u}}{\partial x}=0$$

$$\frac{\partial f}{\partial {\bf u}}={\bf c } \mbox { is a \lq\lq speed''}$$

then $$\frac{\partial {\bf u }}{\partial t}+{\bf c}\frac{\partial {\bf u}}{\partial x}=0$$, the 1-way wave equation.

A very good source of information on approximating solutions numerically to (135) is LeVeque's book.

We'll do a classic example: Burger's Equation in 1-D

Take

$$\displaystyle \frac{\partial u}{\partial t}+\frac12\frac{\partial... ...artial x}(u^2)=0\Rightarrow \displaystyle \frac{\partial u}{\partial t}+uu_x=0$$

with

$$u(x,0)=g(x)\qquad x\in {\mathbb{R}}^1$$

and assume that $$\int^{\infty}_{-\infty}\vert g(x)\vert^2dx<\infty$$ and $g(x)$ is differentiable.

Now can use analysis at beginning of this section to show that the characteristics corresponding to $g(x)=$ a step function will look like those in Figure 28 and at the step rise we get a rarefaction. Can you come up with a $g(x)$ that would lead to a shock, i.e. a crossing of characteristics?

Figure 28: $g(x)=\theta (x)$, a step, leading to a rarefaction.

There are many techniques for approximating solutions to this and other conservation laws $\Rightarrow$ they are front-tracking techniques shape-preserving, flux-limited, etc. But most are based on solving the Riemann problem locally, i.e., following the characteristics locally.

One such family of methods: GODUNOV-METHODS, which is a mildly dissipative method.

Take $\Delta t_n$ be the step size that goes from $n$ to $n+1$ time level. The time step size is variable.

let $u^0_m=\displaystyle \frac{1}{\Delta x}\int^{(m+\frac12)\Delta x}_{(m-\frac12)\Delta x} g(x)dx$

Suppose all $u^{n}_{m}$ are known, we construct a piecewise constant function $w^{[n]}(\cdot, t_n)$ by letting it equal $u^n_m$ in each interval.

Let $\Pi_m\equiv\left(x_{m-\frac12}\right., x_{m+\frac12}\Big]$ and evaluate the exact solution to the Riemann problem ahead of $t=t_n$.

The idea is to let each interval ${\Pi}_m$ ``propagate'' in the direction determined by its characteristics.

Choose a point $(x,t), t\ge t_n$. There are 3 possibilities

(1)

$\exists ! m$ such that the point is reached by a characteristic emanating from $\Pi_m$. Since characteristics propagate constant values, the solution of the Riemann problem at this is $u^n_m$.

(2)

$\exists ! m$ such that the point is reached by characteristics emanating from the intervals $\Pi_m$ and $\Pi_{m+1}$. In this case, as the 2 intervals ``propagate'' in time, they are separated by a shock. The shock advances along a straight line starting at $\left(m+\displaystyle \frac12\right)\Delta x$ and whose slope is the average slopes in $\Pi_m$ and $\Pi_{m+1}$, i.e. $$\frac12\left.\left(u^n_m+u^n_{m+1}\right]\right.$$.

Let this line be $\rho_m$. The value at $(x,t)$ is $u^n_m$ if $x<\rho_m(t)$ and $u^n_{m+1}$ if $x>\rho_m(t)$.

(3)

Characteristics from more than 2 intervals reach the point $(x,t)$. In this case, cannot assign a value to the point.

Simple geometry demonstrates that (3) which we must avoid occurs for $t>\tilde t>t_n$ is the lowest solution to equation $\rho_m(t)=\rho_{m+1}(t)$ for some $m$.

This can be seen in Figure 29

Figure 29: Piece-wise constant approximation of $g(x)$ and the characteristics $\rho$'s emanating from such a $g(x)$, pictured above $g(x)$.

let $t'$ be the time of an encounter between the first encounter. Choose

$$t_{n+1}\in (t_n, t')$$$$\mbox {and } \Delta t_n=t_{n+1}-t_n$$

Since $t_{n+1}\in \left.(t_n, t'\right.\big]$ cases (132)and (133) can be used to construct a $$ solution $w^{[n]}(x,t)\forall t_n\le t\le t_{n+1}$. Choose the

$$u^{n+1}_m=\frac{1}{\Delta x}\int^{(m+\frac12)\Delta x}_{(m-\frac12)\Delta x} w^{[n]} (x,t_{n+1})dx$$ (135)

This integral can be calculated.

Disregarding shocks $w^{[n]}$ obeys Burger's Equation for $t\in[t_n, t_{n+1}]\therefore$

$$\frac{\partial w^{[n]}}{\partial t}+\frac12\frac{\partial (w^{[n]... ...)=w^{[n]}(x,t_n)-\frac12\int^{t_{n+1}}_{t_n} \frac{\partial }{dx}[w^{[n]}]^2dt$$

substituting (136) results in

$$u^{n+1}_m=\frac{1}{\Delta x}\int^{(m+\frac12)\Delta x} _{m+\frac1... ...\frac12\int^{t_{n+1}}_{t_n}\frac{\partial }{\partial x}[w^{[n]}]^2 dt\right\}dx$$

Since $Du^n_{n+m}$ has been obtained by an averaging procedure given in we have after exchanging the order of integration

$$u^{n+1}_{m} = u^n_{m}-\frac{1}{2\Delta x}\int^{t_{n+1}}_{t_n}\int^{(m+\frac12)\Delta x}_{(m-\frac12)\Delta x}\frac{\partial }{\partial x}[w^{[n]}]^2dx dt$$

$$= u^n_m-\frac{1}{2\Delta x}\int^{t_{n+1}}_{t_n} \Big\{\Big[w^{[n]}\... ...ta x, t\Big) \Big]^2-[w^{[n]}\Big(m-\frac12\Big)\Delta x, t\Big)\Big]^2\Big\}dt$$

Recall our definition of $t_{n+1}$. No vertical line segments $\left((m+\displaystyle \frac12)\Delta x, t\right), t\in[t_n, t_{n+1}]$, may cross the discontinuties $\rho_j \therefore$ the value of $w^{[n]}$ across each such segment is constant - equaling $u^n_m$ or $u^n_{m+1}$ (depending on the slope of $\rho_m$: if it points rightwards it is $u^n_m$. Otherwise $u^n_{m+1}$

Denote this value by $\chi_{m+\frac12}$ then

$$u^{n+1}_m=u^n_{m}-\frac12\frac{\Delta t_n}{\Delta x}\left(\chi^2_{m+\frac12}-\chi^2_{m-\frac12}\right)$$

This is the simplest, first order Godunov scheme. $\Box$

In general methods for the approximation of conservation laws should follow characteristics...solve the Riemann problem locally. Godunov is about the local determination of the upwind direction. Another popular upwinding technique is the ENO switch: (see Osher and Engquist) (which is a member of a family of nonlinear techniques that are known as Total Variation Diminishing (TVD) schemes and are very effective in modeling shocks since they have little or no ringing at discontinuities:

$$f_{-(y)}\equiv [\min(y,0)]^2\quad f_{+(y)}\equiv [\max (y,0)]^2$$

$$y\in {\mathbb{R}}$$

to get $$\frac{\partial u}{\partial t}+\frac{1}{\Delta x}\big[\Delta_+f_{-}(u_m)+ \Delta_{-}f_{+}(u_m)\big]=0$$

if $u_{m-1}, u_m, u_{m+1}> 0\rightarrow$ characteristic propagate right $\Rightarrow$

$$\Delta_+f_{-}(u_m)=0$$$$\mbox { and } \Delta_{-}f_{+}(u_m)=[u_m]^2-[u_{m-1}]^2$$

if $u_{m-1}, u_m, u_{m+1}< 0\rightarrow$ characteristic propagate left $\Rightarrow$

$$\Delta_{+}f_{-}(u_m)=[u_{m+1}]^2-[u_m]^2$$$$\quad \mbox {and }\Delta_-f_{+}(u_m)=0.$$

Again, scheme determines upwind direction locally.